Dense matrices using a NumPy backend#
This serves as a base class for dense matrices over Real Double Field and Complex Double Field.
AUTHORS:
Jason Grout, Sep 2008: switch to NumPy backend, factored out the Matrix_double_dense class
Josh Kantor
William Stein: many bug fixes and touch ups.
EXAMPLES:
sage: b = Mat(RDF,2,3).basis()
sage: b[0,0]
[1.0 0.0 0.0]
[0.0 0.0 0.0]
We deal with the case of zero rows or zero columns:
sage: m = MatrixSpace(RDF,0,3)
sage: m.zero_matrix()
[]
- class sage.matrix.matrix_double_dense.Matrix_double_dense#
Bases:
Matrix_numpy_dense
Base class for matrices over the Real Double Field and the Complex Double Field. These are supposed to be fast matrix operations using C doubles. Most operations are implemented using numpy which will call the underlying BLAS on the system.
This class cannot be instantiated on its own. The numpy matrix creation depends on several variables that are set in the subclasses.
EXAMPLES:
sage: m = Matrix(RDF, [[1,2],[3,4]]) sage: m**2 [ 7.0 10.0] [15.0 22.0] sage: m^(-1) # rel tol 1e-15 # needs scipy [-1.9999999999999996 0.9999999999999998] [ 1.4999999999999998 -0.4999999999999999]
- LU()#
Return a decomposition of the (row-permuted) matrix as a product of a lower-triangular matrix (“L”) and an upper-triangular matrix (“U”).
OUTPUT:
For an \(m\times n\) matrix
A
this method returns a triple of immutable matricesP, L, U
such thatA = P*L*U
P
is a square permutation matrix, of size \(m\times m\), so is all zeroes, but with exactly a single one in each row and each column.L
is lower-triangular, square of size \(m\times m\), with every diagonal entry equal to one.U
is upper-triangular with size \(m\times n\), i.e. entries below the “diagonal” are all zero.
The computed decomposition is cached and returned on subsequent calls, thus requiring the results to be immutable.
Effectively,
P
permutes the rows ofA
. ThenL
can be viewed as a sequence of row operations on this matrix, where each operation is adding a multiple of a row to a subsequent row. There is no scaling (thus 1’s on the diagonal ofL
) and no row-swapping (P
does that). As a resultU
is close to being the result of Gaussian-elimination. However, round-off errors can make it hard to determine the zero entries ofU
.Note
The behaviour of
LU()
has changed in Sage version 9.1. Earlier,LU()
returnedP,L,U
such thatP*A=L*U
, whereP
represents the permutation and is the matrix inverse of theP
returned by this method. The computation of this matrix inverse can be accomplished quickly with just a transpose as the matrix is orthogonal/unitary.For details see github issue #18365.
EXAMPLES:
sage: m = matrix(RDF,4,range(16)) sage: P,L,U = m.LU() sage: P*L*U # rel tol 2e-16 [ 0.0 1.0 2.0 3.0] [ 4.0 5.0 6.0 7.0] [ 8.0 9.0 10.0 11.0] [12.0 13.0 14.0 15.0]
Below example illustrates the change in behaviour of
LU()
.sage: (m - P*L*U).norm() < 1e-14 True sage: (P*m - L*U).norm() < 1e-14 False
github issue #10839 made this routine available for rectangular matrices.
sage: A = matrix(RDF, 5, 6, range(30)); A [ 0.0 1.0 2.0 3.0 4.0 5.0] [ 6.0 7.0 8.0 9.0 10.0 11.0] [12.0 13.0 14.0 15.0 16.0 17.0] [18.0 19.0 20.0 21.0 22.0 23.0] [24.0 25.0 26.0 27.0 28.0 29.0] sage: P, L, U = A.LU() sage: P [0.0 1.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 1.0] [0.0 0.0 1.0 0.0 0.0] [0.0 0.0 0.0 1.0 0.0] [1.0 0.0 0.0 0.0 0.0] sage: L.zero_at(0) # Use zero_at(0) to get rid of signed zeros [ 1.0 0.0 0.0 0.0 0.0] [ 0.0 1.0 0.0 0.0 0.0] [ 0.5 0.5 1.0 0.0 0.0] [0.75 0.25 0.0 1.0 0.0] [0.25 0.75 0.0 0.0 1.0] sage: U.zero_at(0) # Use zero_at(0) to get rid of signed zeros [24.0 25.0 26.0 27.0 28.0 29.0] [ 0.0 1.0 2.0 3.0 4.0 5.0] [ 0.0 0.0 0.0 0.0 0.0 0.0] [ 0.0 0.0 0.0 0.0 0.0 0.0] [ 0.0 0.0 0.0 0.0 0.0 0.0] sage: P.transpose()*A-L*U [0.0 0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0 0.0] sage: P*L*U [ 0.0 1.0 2.0 3.0 4.0 5.0] [ 6.0 7.0 8.0 9.0 10.0 11.0] [12.0 13.0 14.0 15.0 16.0 17.0] [18.0 19.0 20.0 21.0 22.0 23.0] [24.0 25.0 26.0 27.0 28.0 29.0]
Trivial cases return matrices of the right size and characteristics.
sage: A = matrix(RDF, 5, 0) sage: P, L, U = A.LU() sage: P.parent() Full MatrixSpace of 5 by 5 dense matrices over Real Double Field sage: L.parent() Full MatrixSpace of 5 by 5 dense matrices over Real Double Field sage: U.parent() Full MatrixSpace of 5 by 0 dense matrices over Real Double Field sage: A-P*L*U []
The results are immutable since they are cached.
sage: P, L, U = matrix(RDF, 2, 2, range(4)).LU() sage: L[0,0] = 0 Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M). sage: P[0,0] = 0 Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M). sage: U[0,0] = 0 Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
- LU_valid()#
Return
True
if the LU form of this matrix has already been computed.EXAMPLES:
sage: A = random_matrix(RDF,3) ; A.LU_valid() False sage: P, L, U = A.LU() sage: A.LU_valid() True
- QR()#
Return a factorization into a unitary matrix and an upper-triangular matrix.
INPUT:
Any matrix over
RDF
orCDF
.OUTPUT:
Q
,R
– a pair of matrices such that if \(A\) is the original matrix, then\[A = QR, \quad Q^\ast Q = I\]where \(R\) is upper-triangular. \(Q^\ast\) is the conjugate-transpose in the complex case, and just the transpose in the real case. So \(Q\) is a unitary matrix (or rather, orthogonal, in the real case), or equivalently \(Q\) has orthogonal columns. For a matrix of full rank this factorization is unique up to adjustments via multiples of rows and columns by multiples with scalars having modulus \(1\). So in the full-rank case, \(R\) is unique if the diagonal entries are required to be positive real numbers.
The resulting decomposition is cached.
ALGORITHM:
Calls “linalg.qr” from SciPy, which is in turn an interface to LAPACK routines.
EXAMPLES:
Over the reals, the inverse of
Q
is its transpose, since including a conjugate has no effect. In the real case, we sayQ
is orthogonal.sage: A = matrix(RDF, [[-2, 0, -4, -1, -1], ....: [-2, 1, -6, -3, -1], ....: [1, 1, 7, 4, 5], ....: [3, 0, 8, 3, 3], ....: [-1, 1, -6, -6, 5]]) sage: Q, R = A.QR()
At this point,
Q
is only well-defined up to the signs of its columns, and similarly forR
and its rows, so we normalize them:sage: Qnorm = Q._normalize_columns() sage: Rnorm = R._normalize_rows() sage: Qnorm.round(6).zero_at(10^-6) [ 0.458831 0.126051 0.381212 0.394574 0.68744] [ 0.458831 -0.47269 -0.051983 -0.717294 0.220963] [-0.229416 -0.661766 0.661923 0.180872 -0.196411] [-0.688247 -0.189076 -0.204468 -0.09663 0.662889] [ 0.229416 -0.535715 -0.609939 0.536422 -0.024551] sage: Rnorm.round(6).zero_at(10^-6) [ 4.358899 -0.458831 13.076697 6.194225 2.982405] [ 0.0 1.670172 0.598741 -1.29202 6.207997] [ 0.0 0.0 5.444402 5.468661 -0.682716] [ 0.0 0.0 0.0 1.027626 -3.6193] [ 0.0 0.0 0.0 0.0 0.024551] sage: (Q*Q.transpose()) # tol 1e-14 [0.9999999999999994 0.0 0.0 0.0 0.0] [ 0.0 1.0 0.0 0.0 0.0] [ 0.0 0.0 0.9999999999999999 0.0 0.0] [ 0.0 0.0 0.0 0.9999999999999998 0.0] [ 0.0 0.0 0.0 0.0 1.0000000000000002] sage: (Q*R - A).zero_at(10^-14) [0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0 0.0]
Now over the complex numbers, demonstrating that the SciPy libraries are (properly) using the Hermitian inner product, so that
Q
is a unitary matrix (its inverse is the conjugate-transpose).sage: A = matrix(CDF, [[-8, 4*I + 1, -I + 2, 2*I + 1], ....: [1, -2*I - 1, -I + 3, -I + 1], ....: [I + 7, 2*I + 1, -2*I + 7, -I + 1], ....: [I + 2, 0, I + 12, -1]]) sage: Q, R = A.QR() sage: Q._normalize_columns() # tol 1e-6 [ 0.7302967433402214 0.20705664550556482 + 0.5383472783144685*I 0.24630498099986423 - 0.07644563587232917*I 0.23816176831943323 - 0.10365960327796941*I] [ -0.09128709291752768 -0.20705664550556482 - 0.37787837804765584*I 0.37865595338630315 - 0.19522214955246678*I 0.7012444502144682 - 0.36437116509865947*I] [ -0.6390096504226938 - 0.09128709291752768*I 0.17082173254209104 + 0.6677576817554466*I -0.03411475806452064 + 0.040901987417671426*I 0.31401710855067644 - 0.08251917187054114*I] [ -0.18257418583505536 - 0.09128709291752768*I -0.03623491296347384 + 0.07246982592694771*I 0.8632284069415112 + 0.06322839976356195*I -0.44996948676115206 - 0.01161191812089182*I] sage: R._normalize_rows().zero_at(1e-15) # tol 1e-6 [ 10.954451150103322 -1.9170289512680814*I 5.385938482134133 - 2.1908902300206643*I -0.2738612787525829 - 2.1908902300206643*I] [ 0.0 4.8295962564173 -0.8696379111233719 - 5.864879483945123*I 0.993871898426711 - 0.30540855212070794*I] [ 0.0 0.0 12.00160760935814 -0.2709533402297273 + 0.4420629644486325*I] [ 0.0 0.0 0.0 1.9429639442589917] sage: (Q.conjugate().transpose()*Q).zero_at(1e-15) # tol 1e-15 [ 1.0 0.0 0.0 0.0] [ 0.0 0.9999999999999994 0.0 0.0] [ 0.0 0.0 1.0000000000000002 0.0] [ 0.0 0.0 0.0 1.0000000000000004] sage: (Q*R - A).zero_at(10^-14) [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0]
An example of a rectangular matrix that is also rank-deficient. If you run this example yourself, you may see a very small, nonzero entries in the third row, in the third column, even though the exact version of the matrix has rank 2. The final two columns of
Q
span the left kernel ofA
(as evidenced by the two zero rows ofR
). Different platforms will compute different bases for this left kernel, so we do not exhibit the actual matrix.sage: Arat = matrix(QQ, [[2, -3, 3], ....: [-1, 1, -1], ....: [-1, 3, -3], ....: [-5, 1, -1]]) sage: Arat.rank() 2 sage: A = Arat.change_ring(CDF) sage: Q, R = A.QR() sage: R._normalize_rows() # abs tol 1e-14 [ 5.567764362830022 -2.6940795304016243 2.6940795304016243] [ 0.0 3.5695847775155825 -3.5695847775155825] [ 0.0 0.0 2.4444034681064287e-16] [ 0.0 0.0 0.0] sage: (Q.conjugate_transpose()*Q) # abs tol 1e-14 [ 1.0000000000000002 -5.185196889911925e-17 -4.1457180570414476e-17 -2.909388767229071e-17] [ -5.185196889911925e-17 1.0000000000000002 -9.286869233696149e-17 -1.1035822863186828e-16] [-4.1457180570414476e-17 -9.286869233696149e-17 1.0 4.4159215672155694e-17] [ -2.909388767229071e-17 -1.1035822863186828e-16 4.4159215672155694e-17 1.0]
Results are cached, meaning they are immutable matrices. Make a copy if you need to manipulate a result.
sage: A = random_matrix(CDF, 2, 2) sage: Q, R = A.QR() sage: Q.is_mutable() False sage: R.is_mutable() False sage: Q[0,0] = 0 Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M). sage: Qcopy = copy(Q) sage: Qcopy[0,0] = 679 sage: Qcopy[0,0] 679.0
- SVD()#
Return the singular value decomposition of this matrix.
The U and V matrices are not unique and may be returned with different values in the future or on different systems. The S matrix is unique and contains the singular values in descending order.
The computed decomposition is cached and returned on subsequent calls.
INPUT:
A – a matrix
OUTPUT:
U, S, V – immutable matrices such that \(A = U*S*V.conj().transpose()\) where U and V are orthogonal and S is zero off of the diagonal.
Note that if self is m-by-n, then the dimensions of the matrices that this returns are (m,m), (m,n), and (n, n).
Note
If all you need is the singular values of the matrix, see the more convenient
singular_values()
.EXAMPLES:
sage: m = matrix(RDF,4,range(1,17)) sage: U,S,V = m.SVD() sage: U*S*V.transpose() # tol 1e-14 [0.9999999999999993 1.9999999999999987 3.000000000000001 4.000000000000002] [ 4.999999999999998 5.999999999999998 6.999999999999998 8.0] [ 8.999999999999998 9.999999999999996 10.999999999999998 12.0] [12.999999999999998 14.0 15.0 16.0]
A non-square example:
sage: m = matrix(RDF, 2, range(1,7)); m [1.0 2.0 3.0] [4.0 5.0 6.0] sage: U, S, V = m.SVD() sage: U*S*V.transpose() # tol 1e-14 [0.9999999999999994 1.9999999999999998 2.999999999999999] [ 4.000000000000001 5.000000000000002 6.000000000000001]
S contains the singular values:
sage: S.round(4) [ 9.508 0.0 0.0] [ 0.0 0.7729 0.0] sage: [N(sqrt(abs(x)), digits=4) for x in (S*S.transpose()).eigenvalues()] [9.508, 0.7729]
U and V are orthogonal matrices:
sage: U # random, SVD is not unique [-0.386317703119 -0.922365780077] [-0.922365780077 0.386317703119] [-0.274721127897 -0.961523947641] [-0.961523947641 0.274721127897] sage: (U*U.transpose()) # tol 1e-15 [ 1.0 0.0] [ 0.0 1.0000000000000004] sage: V # random, SVD is not unique [-0.428667133549 0.805963908589 0.408248290464] [-0.566306918848 0.112382414097 -0.816496580928] [-0.703946704147 -0.581199080396 0.408248290464] sage: (V*V.transpose()) # tol 1e-15 [0.9999999999999999 0.0 0.0] [ 0.0 1.0 0.0] [ 0.0 0.0 0.9999999999999999]
- cholesky()#
Return the Cholesky factorization of a matrix that is real symmetric, or complex Hermitian.
INPUT:
Any square matrix with entries from
RDF
that is symmetric, or with entries fromCDF
that is Hermitian. The matrix must be positive definite for the Cholesky decomposition to exist.OUTPUT:
For a matrix \(A\) the routine returns a lower triangular matrix \(L\) such that,
\[A = LL^\ast\]where \(L^\ast\) is the conjugate-transpose in the complex case, and just the transpose in the real case. If the matrix fails to be positive definite (perhaps because it is not symmetric or Hermitian), then this function raises a
ValueError
.IMPLEMENTATION:
The existence of a Cholesky decomposition and the positive definite property are equivalent. So this method and the
is_positive_definite()
method compute and cache both the Cholesky decomposition and the positive-definiteness. So theis_positive_definite()
method or catching aValueError
from thecholesky()
method are equally expensive computationally and if the decomposition exists, it is cached as a side-effect of either routine.EXAMPLES:
A real matrix that is symmetric, Hermitian, and positive definite:
sage: M = matrix(RDF,[[ 1, 1, 1, 1, 1], ....: [ 1, 5, 31, 121, 341], ....: [ 1, 31, 341, 1555, 4681], ....: [ 1,121, 1555, 7381, 22621], ....: [ 1,341, 4681, 22621, 69905]]) sage: M.is_symmetric() True sage: M.is_hermitian() True sage: L = M.cholesky() sage: L.round(6).zero_at(10^-10) [ 1.0 0.0 0.0 0.0 0.0] [ 1.0 2.0 0.0 0.0 0.0] [ 1.0 15.0 10.723805 0.0 0.0] [ 1.0 60.0 60.985814 7.792973 0.0] [ 1.0 170.0 198.623524 39.366567 1.7231] sage: (L*L.transpose()).round(6).zero_at(10^-10) [ 1.0 1.0 1.0 1.0 1.0] [ 1.0 5.0 31.0 121.0 341.0] [ 1.0 31.0 341.0 1555.0 4681.0] [ 1.0 121.0 1555.0 7381.0 22621.0] [ 1.0 341.0 4681.0 22621.0 69905.0]
A complex matrix that is Hermitian and positive definite.
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 23, 17*I + 3, 24*I + 25, 21*I], ....: [ -17*I + 3, 38, -69*I + 89, 7*I + 15], ....: [-24*I + 25, 69*I + 89, 976, 24*I + 6], ....: [ -21*I, -7*I + 15, -24*I + 6, 28]]) sage: A.is_hermitian() True sage: L = A.cholesky() sage: L.round(6).zero_at(10^-10) [ 4.795832 0.0 0.0 0.0] [ 0.625543 - 3.544745*I 5.004346 0.0 0.0] [ 5.21286 - 5.004346*I 13.588189 + 10.721116*I 24.984023 0.0] [ -4.378803*I -0.104257 - 0.851434*I -0.21486 + 0.371348*I 2.811799] sage: (L*L.conjugate_transpose()).round(6).zero_at(10^-10) [ 23.0 3.0 + 17.0*I 25.0 + 24.0*I 21.0*I] [ 3.0 - 17.0*I 38.0 89.0 - 69.0*I 15.0 + 7.0*I] [25.0 - 24.0*I 89.0 + 69.0*I 976.0 6.0 + 24.0*I] [ -21.0*I 15.0 - 7.0*I 6.0 - 24.0*I 28.0]
This routine will recognize when the input matrix is not positive definite. The negative eigenvalues are an equivalent indicator. (Eigenvalues of a Hermitian matrix must be real, so there is no loss in ignoring the imprecise imaginary parts).
sage: A = matrix(RDF, [[ 3, -6, 9, 6, -9], ....: [-6, 11, -16, -11, 17], ....: [ 9, -16, 28, 16, -40], ....: [ 6, -11, 16, 9, -19], ....: [-9, 17, -40, -19, 68]]) sage: A.is_symmetric() True sage: A.eigenvalues() [108.07..., 13.02..., -0.02..., -0.70..., -1.37...] sage: A.cholesky() Traceback (most recent call last): ... ValueError: matrix is not positive definite sage: # needs sage.symbolic sage: B = matrix(CDF, [[ 2, 4 - 2*I, 2 + 2*I], ....: [4 + 2*I, 8, 10*I], ....: [2 - 2*I, -10*I, -3]]) sage: B.is_hermitian() True sage: [ev.real() for ev in B.eigenvalues()] [15.88..., 0.08..., -8.97...] sage: B.cholesky() Traceback (most recent call last): ... ValueError: matrix is not positive definite
- condition(p='frob')#
Return the condition number of a square nonsingular matrix.
Roughly speaking, this is a measure of how sensitive the matrix is to round-off errors in numerical computations. The minimum possible value is 1.0, and larger numbers indicate greater sensitivity.
INPUT:
p
- default: ‘frob’ - controls which norm is used to compute the condition number, allowable values are ‘frob’ (for the Frobenius norm), integers -2, -1, 1, 2, positive and negative infinity. See output discussion for specifics.
OUTPUT:
The condition number of a matrix is the product of a norm of the matrix times the norm of the inverse of the matrix. This requires that the matrix be square and invertible (nonsingular, full rank).
Returned value is a double precision floating point value in
RDF
, orInfinity
. Row and column sums described below are sums of the absolute values of the entries, where the absolute value of the complex number \(a+bi\) is \(\sqrt{a^2+b^2}\). Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition.p = 'frob'
: the default norm employed in computing the condition number, the Frobenius norm, which for a matrix \(A=(a_{ij})\) computes\[\left(\sum_{i,j}\left\lvert{a_{i,j}}\right\rvert^2\right)^{1/2}\]p = 'sv'
: the quotient of the maximal and minimal singular value.p = Infinity
orp = oo
: the maximum row sum.p = -Infinity
orp = -oo
: the minimum column sum.p = 1
: the maximum column sum.p = -1
: the minimum column sum.p = 2
: the 2-norm, equal to the maximum singular value.p = -2
: the minimum singular value.
ALGORITHM:
Computation is performed by the
cond()
function of the SciPy/NumPy library.EXAMPLES:
First over the reals.
sage: A = matrix(RDF, 4, [(1/4)*x^3 for x in range(16)]); A [ 0.0 0.25 2.0 6.75] [ 16.0 31.25 54.0 85.75] [ 128.0 182.25 250.0 332.75] [ 432.0 549.25 686.0 843.75] sage: A.condition() 9923.88955... sage: A.condition(p='frob') 9923.88955... sage: A.condition(p=Infinity) # tol 3e-14 22738.50000000045 sage: A.condition(p=-Infinity) # tol 2e-14 17.50000000000028 sage: A.condition(p=1) 12139.21... sage: A.condition(p=-1) # tol 2e-14 550.0000000000093 sage: A.condition(p=2) 9897.8088... sage: A.condition(p=-2) 0.000101032462...
And over the complex numbers.
sage: B = matrix(CDF, 3, [x + x^2*I for x in range(9)]); B [ 0.0 1.0 + 1.0*I 2.0 + 4.0*I] [ 3.0 + 9.0*I 4.0 + 16.0*I 5.0 + 25.0*I] [6.0 + 36.0*I 7.0 + 49.0*I 8.0 + 64.0*I] sage: B.condition() 203.851798... sage: B.condition(p='frob') 203.851798... sage: B.condition(p=Infinity) 369.55630... sage: B.condition(p=-Infinity) 5.46112969... sage: B.condition(p=1) 289.251481... sage: B.condition(p=-1) 20.4566639... sage: B.condition(p=2) 202.653543... sage: B.condition(p=-2) 0.00493453005...
Hilbert matrices are famously ill-conditioned, while an identity matrix can hit the minimum with the right norm.
sage: A = matrix(RDF, 10, [1/(i+j+1) for i in range(10) for j in range(10)]) sage: A.condition() # tol 2e-4 16332197709146.014 sage: id = identity_matrix(CDF, 10) sage: id.condition(p=1) 1.0
Return values are in \(RDF\).
sage: A = matrix(CDF, 2, range(1,5)) sage: A.condition() in RDF True
Rectangular and singular matrices raise errors if p is not ‘sv’.
sage: A = matrix(RDF, 2, 3, range(6)) sage: A.condition() Traceback (most recent call last): ... TypeError: matrix must be square if p is not 'sv', not 2 x 3 sage: A.condition('sv') 7.34... sage: A = matrix(QQ, 5, range(25)) sage: A.is_singular() True sage: B = A.change_ring(CDF) sage: B.condition() +Infinity
Improper values of
p
are caught.sage: A = matrix(CDF, 2, range(1,5)) sage: A.condition(p='bogus') Traceback (most recent call last): ... ValueError: condition number 'p' must be +/- infinity, 'frob', 'sv' or an integer, not bogus sage: A.condition(p=632) Traceback (most recent call last): ... ValueError: condition number integer values of 'p' must be -2, -1, 1 or 2, not 632
- conjugate()#
Return the conjugate of this matrix, i.e. the matrix whose entries are the conjugates of the entries of self.
EXAMPLES:
sage: # needs sage.symbolic sage: A = matrix(CDF, [[1+I, 3-I], [0, 2*I]]) sage: A.conjugate() [1.0 - 1.0*I 3.0 + 1.0*I] [ 0.0 -2.0*I]
There is a shorthand notation:
sage: A.conjugate() == A.C # needs sage.symbolic True
Conjugates work (trivially) for real matrices:
sage: B = matrix.random(RDF, 3) sage: B == B.conjugate() True
- determinant()#
Return the determinant of
self
.ALGORITHM:
Use numpy
EXAMPLES:
sage: m = matrix(RDF,2,range(4)); m.det() -2.0 sage: m = matrix(RDF,0,[]); m.det() 1.0 sage: m = matrix(RDF, 2, range(6)); m.det() Traceback (most recent call last): ... ValueError: self must be a square matrix
- eigenvalues(other=None, algorithm='default', tol=None, homogeneous=False)#
Return a list of ordinary or generalized eigenvalues.
INPUT:
self
- a square matrixother
– a square matrix \(B\) (default:None
) in a generalized eigenvalue problem; ifNone
, an ordinary eigenvalue problem is solved; ifalgorithm
is'symmetric'
or'hermitian'
, \(B\) must be real symmetric or hermitian positive definite, respectivelyalgorithm
- default:'default'
'default'
- applicable to any matrix with double-precision floating point entries. Uses theeigvals()
method from SciPy.'symmetric'
- converts the matrix into a real matrix (i.e. with entries fromRDF
), then applies the algorithm for Hermitian matrices. This algorithm can be significantly faster than the'default'
algorithm.'hermitian'
- uses theeigh()
method from SciPy, which applies only to real symmetric or complex Hermitian matrices. Since Hermitian is defined as a matrix equaling its conjugate-transpose, for a matrix with real entries this property is equivalent to being symmetric. This algorithm can be significantly faster than the'default'
algorithm.
'tol'
– (default:None
); if set to a value other thanNone
, this is interpreted as a small real number used to aid in grouping eigenvalues that are numerically similar, but is ignored whenhomogeneous
is set. See the output description for more information.homogeneous
– boolean (default:False
); ifTrue
, use homogeneous coordinates for the output (seeeigenvectors_right()
for details)
Warning
When using the
'symmetric'
or'hermitian'
algorithms, no check is made on the input matrix, and only the entries below, and on, the main diagonal are employed in the computation.Methods such as
is_symmetric()
andis_hermitian()
could be used to verify this beforehand.OUTPUT:
Default output for a square matrix of size \(n\) is a list of \(n\) eigenvalues from the complex double field,
CDF
. If the'symmetric'
or'hermitian'
algorithms are chosen, the returned eigenvalues are from the real double field,RDF
.If a tolerance is specified, an attempt is made to group eigenvalues that are numerically similar. The return is then a list of pairs, where each pair is an eigenvalue followed by its multiplicity. The eigenvalue reported is the mean of the eigenvalues computed, and these eigenvalues are contained in an interval (or disk) whose radius is less than
5*tol
for \(n < 10,000\) in the worst case.More precisely, for an \(n\times n\) matrix, the diameter of the interval containing similar eigenvalues could be as large as sum of the reciprocals of the first \(n\) integers times
tol
.Warning
Use caution when using the
tol
parameter to group eigenvalues. See the examples below to see how this can go wrong.EXAMPLES:
sage: m = matrix(RDF, 2, 2, [1,2,3,4]) sage: ev = m.eigenvalues(); ev [-0.372281323..., 5.37228132...] sage: ev[0].parent() Complex Double Field sage: m = matrix(RDF, 2, 2, [0,1,-1,0]) sage: m.eigenvalues(algorithm='default') [1.0*I, -1.0*I] sage: m = matrix(CDF, 2, 2, [I,1,-I,0]) # needs sage.symbolic sage: m.eigenvalues() # needs sage.symbolic [-0.624810533... + 1.30024259...*I, 0.624810533... - 0.30024259...*I]
The adjacency matrix of a graph will be symmetric, and the eigenvalues will be real.
sage: # needs sage.graphs sage: A = graphs.PetersenGraph().adjacency_matrix() sage: A = A.change_ring(RDF) sage: ev = A.eigenvalues(algorithm='symmetric'); ev # tol 1e-14 [-2.0, -2.0, -2.0, -2.0, 1.0, 1.0, 1.0, 1.0, 1.0, 3.0] sage: ev[0].parent() Real Double Field
The matrix
A
is “random”, but the construction ofC
provides a positive-definite Hermitian matrix. Note that the eigenvalues of a Hermitian matrix are real, and the eigenvalues of a positive-definite matrix will be positive.sage: # needs sage.symbolic sage: A = matrix([[ 4*I + 5, 8*I + 1, 7*I + 5, 3*I + 5], ....: [ 7*I - 2, -4*I + 7, -2*I + 4, 8*I + 8], ....: [-2*I + 1, 6*I + 6, 5*I + 5, -I - 4], ....: [ 5*I + 1, 6*I + 2, I - 4, -I + 3]]) sage: C = (A*A.conjugate_transpose()).change_ring(CDF) sage: ev = C.eigenvalues(algorithm='hermitian'); ev [2.68144025..., 49.5167998..., 274.086188..., 390.71557...] sage: ev[0].parent() Real Double Field
A tolerance can be given to aid in grouping eigenvalues that are similar numerically. However, if the parameter is too small it might split too finely. Too large, and it can go wrong very badly. Use with care.
sage: # needs sage.graphs sage: G = graphs.PetersenGraph() sage: G.spectrum() [3, 1, 1, 1, 1, 1, -2, -2, -2, -2] sage: A = G.adjacency_matrix().change_ring(RDF) sage: A.eigenvalues(algorithm='symmetric', tol=1.0e-5) # tol 1e-15 [(-2.0, 4), (1.0, 5), (3.0, 1)] sage: A.eigenvalues(algorithm='symmetric', tol=2.5) # tol 1e-15 [(-2.0, 4), (1.3333333333333333, 6)]
An (extreme) example of properly grouping similar eigenvalues.
sage: # needs sage.graphs sage: G = graphs.HigmanSimsGraph() sage: A = G.adjacency_matrix().change_ring(RDF) sage: A.eigenvalues(algorithm='symmetric', tol=1.0e-5) # tol 2e-15 [(-8.0, 22), (2.0, 77), (22.0, 1)]
In this generalized eigenvalue problem, the homogeneous coordinates explain the output obtained for the eigenvalues:
sage: A = matrix.identity(RDF, 2) sage: B = matrix(RDF, [[3, 5], [6, 10]]) sage: A.eigenvalues(B) # tol 1e-14 [0.0769230769230769, +infinity] sage: E = A.eigenvalues(B, homogeneous=True); E # random [(0.9999999999999999, 13.000000000000002), (0.9999999999999999, 0.0)] sage: [alpha/beta for alpha, beta in E] # tol 1e-14 [0.0769230769230769, NaN + NaN*I]
- eigenvectors_left(other=None, homogeneous=False)#
Compute the ordinary or generalized left eigenvectors of a matrix of double precision real or complex numbers (i.e.
RDF
orCDF
).INPUT:
other
– a square matrix \(B\) (default:None
) in a generalized eigenvalue problem; ifNone
, an ordinary eigenvalue problem is solvedhomogeneous
– boolean (default:False
); ifTrue
, use homogeneous coordinates for the eigenvalues in the output
OUTPUT:
A list of triples, each of the form
(e,[v],1)
, wheree
is the eigenvalue, andv
is an associated left eigenvector such that\[v A = e v.\]If the matrix \(A\) is of size \(n\), then there are \(n\) triples.
If a matrix \(B\) is passed as optional argument, the output is a solution to the generalized eigenvalue problem such that
\[v A = e v B.\]If
homogeneous
is set, each eigenvalue is returned as a tuple \((\alpha, \beta)\) of homogeneous coordinates such that\[\beta v A = \alpha v B.\]The format of the output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a complex sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
ALGORITHM:
Values are computed with the SciPy library using
scipy:scipy.linalg.eig()
.EXAMPLES:
sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]]) sage: m [ -5.0 3.0 2.0 8.0] [ 10.0 2.0 4.0 -2.0] [ -1.0 -10.0 -10.0 -17.0] [ -2.0 7.0 6.0 13.0] sage: spectrum = m.left_eigenvectors() sage: for i in range(len(spectrum)): ....: spectrum[i][1][0] = matrix(RDF, spectrum[i][1]).echelon_form()[0] sage: spectrum[0] # tol 1e-13 (2.0, [(1.0, 1.0, 1.0, 1.0)], 1) sage: spectrum[1] # tol 1e-13 (1.0, [(1.0, 0.8, 0.8, 0.6)], 1) sage: spectrum[2] # tol 1e-13 (-2.0, [(1.0, 0.4, 0.6, 0.2)], 1) sage: spectrum[3] # tol 1e-13 (-1.0, [(1.0, 1.0, 2.0, 2.0)], 1)
A generalized eigenvalue problem:
sage: A = matrix(CDF, [[1+I, -2], [3, 4]]) sage: B = matrix(CDF, [[0, 7-I], [2, -3]]) sage: E = A.eigenvectors_left(B) sage: all((v * A - e * v * B).norm() < 1e-14 for e, [v], _ in E) True
In a generalized eigenvalue problem with a singular matrix \(B\), we can check the eigenvector property using homogeneous coordinates, even though the quotient \(\alpha/\beta\) is not always defined:
sage: A = matrix.identity(CDF, 2) sage: B = matrix(CDF, [[2, 1+I], [4, 2+2*I]]) sage: E = A.eigenvectors_left(B, homogeneous=True) sage: all((beta * v * A - alpha * v * B).norm() < 1e-14 ....: for (alpha, beta), [v], _ in E) True
- eigenvectors_right(other=None, homogeneous=False)#
Compute the ordinary or generalized right eigenvectors of a matrix of double precision real or complex numbers (i.e.
RDF
orCDF
).INPUT:
other
– a square matrix \(B\) (default:None
) in a generalized eigenvalue problem; ifNone
, an ordinary eigenvalue problem is solvedhomogeneous
– boolean (default:False
); ifTrue
, use homogeneous coordinates for the eigenvalues in the output
OUTPUT:
A list of triples, each of the form
(e,[v],1)
, wheree
is the eigenvalue, andv
is an associated right eigenvector such that\[A v = e v.\]If the matrix \(A\) is of size \(n\), then there are \(n\) triples.
If a matrix \(B\) is passed as optional argument, the output is a solution to the generalized eigenvalue problem such that
\[A v = e B v.\]If
homogeneous
is set, each eigenvalue is returned as a tuple \((\alpha, \beta)\) of homogeneous coordinates such that\[\beta A v = \alpha B v.\]The format of the output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a complex sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
ALGORITHM:
Values are computed with the SciPy library using
scipy:scipy.linalg.eig()
.EXAMPLES:
sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]]) sage: m [ -9.0 -14.0 19.0 -74.0] [ -1.0 2.0 4.0 -11.0] [ -4.0 -12.0 6.0 -32.0] [ 0.0 -2.0 -1.0 1.0] sage: spectrum = m.right_eigenvectors() sage: for i in range(len(spectrum)): ....: spectrum[i][1][0] = matrix(RDF, spectrum[i][1]).echelon_form()[0] sage: spectrum[0] # tol 1e-13 (2.0, [(1.0, -2.0, 3.0, 1.0)], 1) sage: spectrum[1] # tol 1e-13 (1.0, [(1.0, -0.666666666666633, 1.333333333333286, 0.33333333333331555)], 1) sage: spectrum[2] # tol 1e-13 (-2.0, [(1.0, -0.2, 1.0, 0.2)], 1) sage: spectrum[3] # tol 1e-13 (-1.0, [(1.0, -0.5, 2.0, 0.5)], 1)
A generalized eigenvalue problem:
sage: A = matrix(CDF, [[1+I, -2], [3, 4]]) sage: B = matrix(CDF, [[0, 7-I], [2, -3]]) sage: E = A.eigenvectors_right(B) sage: all((A * v - e * B * v).norm() < 1e-14 for e, [v], _ in E) True
In a generalized eigenvalue problem with a singular matrix \(B\), we can check the eigenvector property using homogeneous coordinates, even though the quotient \(\alpha/\beta\) is not always defined:
sage: A = matrix.identity(RDF, 2) sage: B = matrix(RDF, [[3, 5], [6, 10]]) sage: E = A.eigenvectors_right(B, homogeneous=True) sage: all((beta * A * v - alpha * B * v).norm() < 1e-14 ....: for (alpha, beta), [v], _ in E) True
- exp()#
Calculate the exponential of this matrix X, which is the matrix
\[e^X = \sum_{k=0}^{\infty} \frac{X^k}{k!}.\]EXAMPLES:
sage: A = matrix(RDF, 2, [1,2,3,4]); A [1.0 2.0] [3.0 4.0] sage: A.exp() # tol 1e-14 [51.968956198705044 74.73656456700327] [112.10484685050491 164.07380304920997] sage: A = matrix(CDF, 2, [1,2+I,3*I,4]); A # needs sage.symbolic [ 1.0 2.0 + 1.0*I] [ 3.0*I 4.0] sage: A.exp() # tol 1.1e-14 # needs sage.symbolic [-19.614602953804912 + 12.517743846762578*I 3.7949636449582176 + 28.88379930658099*I] [ -32.383580980922254 + 21.88423595789845*I 2.269633004093535 + 44.901324827684824*I]
- is_hermitian(tol=1e-12, algorithm='naive')#
Return
True
if the matrix is equal to its conjugate-transpose.INPUT:
tol
- default:1e-12
- the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal.algorithm
– string (default: “naive”); either “naive” or “orthonormal”
OUTPUT:
True
if the matrix is square and equal to the transpose with every entry conjugated, andFalse
otherwise.Note that if conjugation has no effect on elements of the base ring (such as for integers), then the
is_symmetric()
method is equivalent and faster.The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply compares corresponding entries on either side of the diagonal (and on the diagonal itself) to see if they are conjugates, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the
schur()
method) and checks that the result is a diagonal matrix with real entries.So the naive algorithm can finish quickly for a matrix that is not Hermitian, while the orthonormal algorithm will always compute a Schur decomposition before going through a similar check of the matrix entry-by-entry.
EXAMPLES:
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I], ....: [-3 - I, -4*I, -2], ....: [-1 + I, -2 - 8*I, 2 + I]]) sage: A.is_hermitian(algorithm='orthonormal') False sage: A.is_hermitian(algorithm='naive') False sage: B = A*A.conjugate_transpose() sage: B.is_hermitian(algorithm='orthonormal') True sage: B.is_hermitian(algorithm='naive') True
A matrix that is nearly Hermitian, but for one non-real diagonal entry.
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 2, 2-I, 1+4*I], ....: [ 2+I, 3+I, 2-6*I], ....: [1-4*I, 2+6*I, 5]]) sage: A.is_hermitian(algorithm='orthonormal') False sage: A[1,1] = 132 sage: A.is_hermitian(algorithm='orthonormal') True
We get a unitary matrix from the SVD routine and use this numerical matrix to create a matrix that should be Hermitian (indeed it should be the identity matrix), but with some imprecision. We use this to illustrate that if the tolerance is set too small, then we can be too strict about the equality of entries and may achieve the wrong result (depending on the system):
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I], ....: [-3 - I, -4*I, -2], ....: [-1 + I, -2 - 8*I, 2 + I]]) sage: U, _, _ = A.SVD() sage: B = U*U.conjugate_transpose() sage: B.is_hermitian(algorithm='naive') True sage: B.is_hermitian(algorithm='naive', tol=1.0e-17) # random False sage: B.is_hermitian(algorithm='naive', tol=1.0e-15) True
A square, empty matrix is trivially Hermitian.
sage: A = matrix(RDF, 0, 0) sage: A.is_hermitian() True
Rectangular matrices are never Hermitian, no matter which algorithm is requested.
sage: A = matrix(CDF, 3, 4) sage: A.is_hermitian() False
AUTHOR:
Rob Beezer (2011-03-30)
- is_normal(tol=1e-12, algorithm='orthonormal')#
Return
True
if the matrix commutes with its conjugate-transpose.INPUT:
tol
- default:1e-12
- the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal.algorithm
- default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure.
OUTPUT:
True
if the matrix is square and commutes with its conjugate-transpose, andFalse
otherwise.Normal matrices are precisely those that can be diagonalized by a unitary matrix.
The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply compares entries of the two possible products of the matrix with its conjugate-transpose, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the
schur()
method) and checks that the result is a diagonal matrix. An orthonormal diagonalization is equivalent to being normal.So the naive algorithm can finish fairly quickly for a matrix that is not normal, once the products have been computed. However, the orthonormal algorithm will compute a Schur decomposition before going through a similar check of a matrix entry-by-entry.
EXAMPLES:
First over the complexes.
B
is Hermitian, hence normal.sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I], ....: [-3 - I, -4*I, -2], ....: [-1 + I, -2 - 8*I, 2 + I]]) sage: B = A*A.conjugate_transpose() sage: B.is_hermitian() True sage: B.is_normal(algorithm='orthonormal') True sage: B.is_normal(algorithm='naive') True sage: B[0,0] = I sage: B.is_normal(algorithm='orthonormal') False sage: B.is_normal(algorithm='naive') False
Now over the reals. Circulant matrices are normal.
sage: # needs sage.graphs sage: G = graphs.CirculantGraph(20, [3, 7]) sage: D = digraphs.Circuit(20) sage: A = 3*D.adjacency_matrix() - 5*G.adjacency_matrix() sage: A = A.change_ring(RDF) sage: A.is_normal() True sage: A.is_normal(algorithm='naive') True sage: A[19,0] = 4.0 sage: A.is_normal() False sage: A.is_normal(algorithm='naive') False
Skew-Hermitian matrices are normal.
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I], ....: [-3 - I, -4*I, -2], ....: [-1 + I, -2 - 8*I, 2 + I]]) sage: B = A - A.conjugate_transpose() sage: B.is_hermitian() False sage: B.is_normal() True sage: B.is_normal(algorithm='naive') True
A small matrix that does not fit into any of the usual categories of normal matrices.
sage: A = matrix(RDF, [[1, -1], ....: [1, 1]]) sage: A.is_normal() True sage: not A.is_hermitian() and not A.is_skew_symmetric() True
Sage has several fields besides the entire complex numbers where conjugation is non-trivial.
sage: # needs sage.rings.number_field sage: F.<b> = QuadraticField(-7) sage: C = matrix(F, [[-2*b - 3, 7*b - 6, -b + 3], ....: [-2*b - 3, -3*b + 2, -2*b], ....: [ b + 1, 0, -2]]) sage: C = C*C.conjugate_transpose() sage: C.is_normal() True
A square, empty matrix is trivially normal.
sage: A = matrix(CDF, 0, 0) sage: A.is_normal() True
Rectangular matrices are never normal, no matter which algorithm is requested.
sage: A = matrix(CDF, 3, 4) sage: A.is_normal() False
AUTHOR:
Rob Beezer (2011-03-31)
- is_positive_definite()#
Determines if a matrix is positive definite.
A matrix \(A\) is positive definite if it is square, is Hermitian (which reduces to symmetric in the real case), and for every nonzero vector \(\vec{x}\),
\[\vec{x}^\ast A \vec{x} > 0\]where \(\vec{x}^\ast\) is the conjugate-transpose in the complex case and just the transpose in the real case. Equivalently, a positive definite matrix has only positive eigenvalues and only positive determinants of leading principal submatrices.
INPUT:
Any matrix over
RDF
orCDF
.OUTPUT:
True
if and only if the matrix is square, Hermitian, and meets the condition above on the quadratic form. The result is cached.IMPLEMENTATION:
The existence of a Cholesky decomposition and the positive definite property are equivalent. So this method and the
cholesky()
method compute and cache both the Cholesky decomposition and the positive-definiteness. So theis_positive_definite()
method or catching aValueError
from thecholesky()
method are equally expensive computationally and if the decomposition exists, it is cached as a side-effect of either routine.EXAMPLES:
A matrix over
RDF
that is positive definite.sage: M = matrix(RDF,[[ 1, 1, 1, 1, 1], ....: [ 1, 5, 31, 121, 341], ....: [ 1, 31, 341, 1555, 4681], ....: [ 1,121, 1555, 7381, 22621], ....: [ 1,341, 4681, 22621, 69905]]) sage: M.is_symmetric() True sage: M.eigenvalues() [77547.66..., 82.44..., 2.41..., 0.46..., 0.011...] sage: [round(M[:i,:i].determinant()) for i in range(1, M.nrows()+1)] [1, 4, 460, 27936, 82944] sage: M.is_positive_definite() True
A matrix over
CDF
that is positive definite.sage: # needs sage.symbolic sage: C = matrix(CDF, [[ 23, 17*I + 3, 24*I + 25, 21*I], ....: [ -17*I + 3, 38, -69*I + 89, 7*I + 15], ....: [-24*I + 25, 69*I + 89, 976, 24*I + 6], ....: [ -21*I, -7*I + 15, -24*I + 6, 28]]) sage: C.is_hermitian() True sage: [x.real() for x in C.eigenvalues()] [991.46..., 55.96..., 3.69..., 13.87...] sage: [round(C[:i,:i].determinant().real()) for i in range(1, C.nrows()+1)] [23, 576, 359540, 2842600] sage: C.is_positive_definite() True
A matrix over
RDF
that is not positive definite.sage: A = matrix(RDF, [[ 3, -6, 9, 6, -9], ....: [-6, 11, -16, -11, 17], ....: [ 9, -16, 28, 16, -40], ....: [ 6, -11, 16, 9, -19], ....: [-9, 17, -40, -19, 68]]) sage: A.is_symmetric() True sage: A.eigenvalues() [108.07..., 13.02..., -0.02..., -0.70..., -1.37...] sage: [round(A[:i,:i].determinant()) for i in range(1, A.nrows()+1)] [3, -3, -15, 30, -30] sage: A.is_positive_definite() False
A matrix over
CDF
that is not positive definite.sage: # needs sage.symbolic sage: B = matrix(CDF, [[ 2, 4 - 2*I, 2 + 2*I], ....: [4 + 2*I, 8, 10*I], ....: [2 - 2*I, -10*I, -3]]) sage: B.is_hermitian() True sage: [ev.real() for ev in B.eigenvalues()] [15.88..., 0.08..., -8.97...] sage: [round(B[:i,:i].determinant().real()) for i in range(1, B.nrows()+1)] [2, -4, -12] sage: B.is_positive_definite() False
A large random matrix that is guaranteed by theory to be positive definite.
sage: R = random_matrix(CDF, 200) sage: H = R.conjugate_transpose()*R sage: H.is_positive_definite() True
AUTHOR:
Rob Beezer (2012-05-28)
- is_skew_hermitian(tol=1e-12, algorithm='orthonormal')#
Return
True
if the matrix is equal to the negative of its conjugate transpose.INPUT:
tol
- default:1e-12
- the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal.algorithm
- default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure.
OUTPUT:
True
if the matrix is square and equal to the negative of its conjugate transpose, andFalse
otherwise.Note that if conjugation has no effect on elements of the base ring (such as for integers), then the
is_skew_symmetric()
method is equivalent and faster.The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply compares corresponding entries on either side of the diagonal (and on the diagonal itself) to see if they are conjugates, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the
schur()
method) and checks that the result is a diagonal matrix with real entries.So the naive algorithm can finish quickly for a matrix that is not Hermitian, while the orthonormal algorithm will always compute a Schur decomposition before going through a similar check of the matrix entry-by-entry.
EXAMPLES:
sage: A = matrix(CDF, [[0, -1], ....: [1, 0]]) sage: A.is_skew_hermitian(algorithm='orthonormal') True sage: A.is_skew_hermitian(algorithm='naive') True
A matrix that is nearly skew-Hermitian, but for a non-real diagonal entry.
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ -I, -1, 1-I], ....: [ 1, 1, -1], ....: [-1-I, 1, -I]]) sage: A.is_skew_hermitian() False sage: A[1,1] = -I sage: A.is_skew_hermitian() True
We get a unitary matrix from the SVD routine and use this numerical matrix to create a matrix that should be skew-Hermitian (indeed it should be the identity matrix multiplied by \(I\)), but with some imprecision. We use this to illustrate that if the tolerance is set too small, then we can be too strict about the equality of entries and may achieve the wrong result (depending on the system):
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I], ....: [-3 - I, -4*I, -2], ....: [-1 + I, -2 - 8*I, 2 + I]]) sage: U, _, _ = A.SVD() sage: B = 1j*U*U.conjugate_transpose() sage: B.is_skew_hermitian(algorithm='naive') True sage: B.is_skew_hermitian(algorithm='naive', tol=1.0e-17) # random False sage: B.is_skew_hermitian(algorithm='naive', tol=1.0e-15) True
A square, empty matrix is trivially Hermitian.
sage: A = matrix(RDF, 0, 0) sage: A.is_skew_hermitian() True
Rectangular matrices are never Hermitian, no matter which algorithm is requested.
sage: A = matrix(CDF, 3, 4) sage: A.is_skew_hermitian() False
AUTHOR:
Rob Beezer (2011-03-30)
- is_unitary(tol=1e-12, algorithm='orthonormal')#
Return
True
if the columns of the matrix are an orthonormal basis.For a matrix with real entries this determines if a matrix is “orthogonal” and for a matrix with complex entries this determines if the matrix is “unitary.”
INPUT:
tol
- default:1e-12
- the largest value of the absolute value of the difference between two matrix entries for which they will still be considered equal.algorithm
- default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’ for a fast procedure.
OUTPUT:
True
if the matrix is square and its conjugate-transpose is its inverse, andFalse
otherwise. In other words, a matrix is orthogonal or unitary if the product of its conjugate-transpose times the matrix is the identity matrix.The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply computes the product of the conjugate-transpose with the matrix and compares the entries to the identity matrix, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the
schur()
method) and checks that the result is a diagonal matrix with entries of modulus 1, which is equivalent to being unitary.So the naive algorithm might finish fairly quickly for a matrix that is not unitary, once the product has been computed. However, the orthonormal algorithm will compute a Schur decomposition before going through a similar check of a matrix entry-by-entry.
EXAMPLES:
A matrix that is far from unitary.
sage: A = matrix(RDF, 4, range(16)) sage: A.conjugate().transpose()*A [224.0 248.0 272.0 296.0] [248.0 276.0 304.0 332.0] [272.0 304.0 336.0 368.0] [296.0 332.0 368.0 404.0] sage: A.is_unitary() False sage: A.is_unitary(algorithm='naive') False sage: A.is_unitary(algorithm='orthonormal') False
The QR decomposition will produce a unitary matrix as Q and the SVD decomposition will create two unitary matrices, U and V.
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 1 - I, -3*I, -2 + I, 1, -2 + 3*I], ....: [ 1 - I, -2 + I, 1 + 4*I, 0, 2 + I], ....: [ -1, -5 + I, -2 + I, 1 + I, -5 - 4*I], ....: [-2 + 4*I, 2 - I, 8 - 4*I, 1 - 8*I, 3 - 2*I]]) sage: Q, R = A.QR() sage: Q.is_unitary() True sage: U, S, V = A.SVD() sage: U.is_unitary(algorithm='naive') True sage: U.is_unitary(algorithm='orthonormal') True sage: V.is_unitary(algorithm='naive') True
If we make the tolerance too strict we can get misleading results.
sage: A = matrix(RDF, 10, 10, [1/(i+j+1) for i in range(10) for j in range(10)]) sage: Q, R = A.QR() sage: Q.is_unitary(algorithm='naive', tol=1e-16) False sage: Q.is_unitary(algorithm='orthonormal', tol=1e-17) False
Rectangular matrices are not unitary/orthogonal, even if their columns form an orthonormal set.
sage: A = matrix(CDF, [[1,0], [0,0], [0,1]]) sage: A.is_unitary() False
The smallest cases:
sage: P = matrix(CDF, 0, 0) sage: P.is_unitary(algorithm='naive') True sage: P = matrix(CDF, 1, 1, [1]) sage: P.is_unitary(algorithm='orthonormal') True sage: P = matrix(CDF, 0, 0,) sage: P.is_unitary(algorithm='orthonormal') True
AUTHOR:
Rob Beezer (2011-05-04)
- left_eigenvectors(other=None, homogeneous=False)#
Compute the ordinary or generalized left eigenvectors of a matrix of double precision real or complex numbers (i.e.
RDF
orCDF
).INPUT:
other
– a square matrix \(B\) (default:None
) in a generalized eigenvalue problem; ifNone
, an ordinary eigenvalue problem is solvedhomogeneous
– boolean (default:False
); ifTrue
, use homogeneous coordinates for the eigenvalues in the output
OUTPUT:
A list of triples, each of the form
(e,[v],1)
, wheree
is the eigenvalue, andv
is an associated left eigenvector such that\[v A = e v.\]If the matrix \(A\) is of size \(n\), then there are \(n\) triples.
If a matrix \(B\) is passed as optional argument, the output is a solution to the generalized eigenvalue problem such that
\[v A = e v B.\]If
homogeneous
is set, each eigenvalue is returned as a tuple \((\alpha, \beta)\) of homogeneous coordinates such that\[\beta v A = \alpha v B.\]The format of the output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a complex sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
ALGORITHM:
Values are computed with the SciPy library using
scipy:scipy.linalg.eig()
.EXAMPLES:
sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]]) sage: m [ -5.0 3.0 2.0 8.0] [ 10.0 2.0 4.0 -2.0] [ -1.0 -10.0 -10.0 -17.0] [ -2.0 7.0 6.0 13.0] sage: spectrum = m.left_eigenvectors() sage: for i in range(len(spectrum)): ....: spectrum[i][1][0] = matrix(RDF, spectrum[i][1]).echelon_form()[0] sage: spectrum[0] # tol 1e-13 (2.0, [(1.0, 1.0, 1.0, 1.0)], 1) sage: spectrum[1] # tol 1e-13 (1.0, [(1.0, 0.8, 0.8, 0.6)], 1) sage: spectrum[2] # tol 1e-13 (-2.0, [(1.0, 0.4, 0.6, 0.2)], 1) sage: spectrum[3] # tol 1e-13 (-1.0, [(1.0, 1.0, 2.0, 2.0)], 1)
A generalized eigenvalue problem:
sage: A = matrix(CDF, [[1+I, -2], [3, 4]]) sage: B = matrix(CDF, [[0, 7-I], [2, -3]]) sage: E = A.eigenvectors_left(B) sage: all((v * A - e * v * B).norm() < 1e-14 for e, [v], _ in E) True
In a generalized eigenvalue problem with a singular matrix \(B\), we can check the eigenvector property using homogeneous coordinates, even though the quotient \(\alpha/\beta\) is not always defined:
sage: A = matrix.identity(CDF, 2) sage: B = matrix(CDF, [[2, 1+I], [4, 2+2*I]]) sage: E = A.eigenvectors_left(B, homogeneous=True) sage: all((beta * v * A - alpha * v * B).norm() < 1e-14 ....: for (alpha, beta), [v], _ in E) True
- log_determinant()#
Compute the log of the absolute value of the determinant using LU decomposition.
Note
This is useful if the usual determinant overflows.
EXAMPLES:
sage: m = matrix(RDF,2,2,range(4)); m [0.0 1.0] [2.0 3.0] sage: RDF(log(abs(m.determinant()))) 0.6931471805599453 sage: m.log_determinant() 0.6931471805599453 sage: m = matrix(RDF,0,0,[]); m [] sage: m.log_determinant() 0.0 sage: m = matrix(CDF,2,2,range(4)); m [0.0 1.0] [2.0 3.0] sage: RDF(log(abs(m.determinant()))) 0.6931471805599453 sage: m.log_determinant() 0.6931471805599453 sage: m = matrix(CDF,0,0,[]); m [] sage: m.log_determinant() 0.0
- norm(p=2)#
Return the norm of the matrix.
INPUT:
p
- default: 2 - controls which norm is computed, allowable values are ‘frob’ (for the Frobenius norm), integers -2, -1, 1, 2, positive and negative infinity. See output discussion for specifics.
OUTPUT:
Returned value is a double precision floating point value in
RDF
. Row and column sums described below are sums of the absolute values of the entries, where the absolute value of the complex number \(a+bi\) is \(\sqrt{a^2+b^2}\). Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition.p = 'frob'
: the Frobenius norm, which for a matrix \(A=(a_{ij})\) computes\[\left(\sum_{i,j}\left\lvert{a_{i,j}}\right\rvert^2\right)^{1/2}\]p = Infinity
orp = oo
: the maximum row sum.p = -Infinity
orp = -oo
: the minimum column sum.p = 1
: the maximum column sum.p = -1
: the minimum column sum.p = 2
: the induced 2-norm, equal to the maximum singular value.p = -2
: the minimum singular value.
ALGORITHM:
Computation is performed by the
norm()
function of the SciPy/NumPy library.EXAMPLES:
First over the reals.
sage: A = matrix(RDF, 3, range(-3, 6)); A [-3.0 -2.0 -1.0] [ 0.0 1.0 2.0] [ 3.0 4.0 5.0] sage: A.norm() 7.99575670... sage: A.norm(p='frob') 8.30662386... sage: A.norm(p=Infinity) 12.0 sage: A.norm(p=-Infinity) 3.0 sage: A.norm(p=1) 8.0 sage: A.norm(p=-1) 6.0 sage: A.norm(p=2) 7.99575670... sage: A.norm(p=-2) < 10^-15 True
And over the complex numbers.
sage: # needs sage.symbolic sage: B = matrix(CDF, 2, [[1+I, 2+3*I],[3+4*I,3*I]]); B [1.0 + 1.0*I 2.0 + 3.0*I] [3.0 + 4.0*I 3.0*I] sage: B.norm() 6.66189877... sage: B.norm(p='frob') 7.0 sage: B.norm(p=Infinity) 8.0 sage: B.norm(p=-Infinity) 5.01976483... sage: B.norm(p=1) 6.60555127... sage: B.norm(p=-1) 6.41421356... sage: B.norm(p=2) 6.66189877... sage: B.norm(p=-2) 2.14921023...
Since it is invariant under unitary multiplication, the Frobenius norm is equal to the square root of the sum of squares of the singular values.
sage: A = matrix(RDF, 5, range(1,26)) sage: f = A.norm(p='frob') sage: U, S, V = A.SVD() sage: s = sqrt(sum([S[i,i]^2 for i in range(5)])) sage: abs(f-s) < 1.0e-12 True
Return values are in \(RDF\).
sage: A = matrix(CDF, 2, range(4)) sage: A.norm() in RDF True
Improper values of
p
are caught.sage: A.norm(p='bogus') Traceback (most recent call last): ... ValueError: matrix norm 'p' must be +/- infinity, 'frob' or an integer, not bogus sage: A.norm(p=632) Traceback (most recent call last): ... ValueError: matrix norm integer values of 'p' must be -2, -1, 1 or 2, not 632
- right_eigenvectors(other=None, homogeneous=False)#
Compute the ordinary or generalized right eigenvectors of a matrix of double precision real or complex numbers (i.e.
RDF
orCDF
).INPUT:
other
– a square matrix \(B\) (default:None
) in a generalized eigenvalue problem; ifNone
, an ordinary eigenvalue problem is solvedhomogeneous
– boolean (default:False
); ifTrue
, use homogeneous coordinates for the eigenvalues in the output
OUTPUT:
A list of triples, each of the form
(e,[v],1)
, wheree
is the eigenvalue, andv
is an associated right eigenvector such that\[A v = e v.\]If the matrix \(A\) is of size \(n\), then there are \(n\) triples.
If a matrix \(B\) is passed as optional argument, the output is a solution to the generalized eigenvalue problem such that
\[A v = e B v.\]If
homogeneous
is set, each eigenvalue is returned as a tuple \((\alpha, \beta)\) of homogeneous coordinates such that\[\beta A v = \alpha B v.\]The format of the output is designed to match the format for exact results. However, since matrices here have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on different hardware they may vary by a complex sign. So for doctests we have normalized output by forcing their eigenvectors to have their first non-zero entry equal to one.
ALGORITHM:
Values are computed with the SciPy library using
scipy:scipy.linalg.eig()
.EXAMPLES:
sage: m = matrix(RDF, [[-9, -14, 19, -74],[-1, 2, 4, -11],[-4, -12, 6, -32],[0, -2, -1, 1]]) sage: m [ -9.0 -14.0 19.0 -74.0] [ -1.0 2.0 4.0 -11.0] [ -4.0 -12.0 6.0 -32.0] [ 0.0 -2.0 -1.0 1.0] sage: spectrum = m.right_eigenvectors() sage: for i in range(len(spectrum)): ....: spectrum[i][1][0] = matrix(RDF, spectrum[i][1]).echelon_form()[0] sage: spectrum[0] # tol 1e-13 (2.0, [(1.0, -2.0, 3.0, 1.0)], 1) sage: spectrum[1] # tol 1e-13 (1.0, [(1.0, -0.666666666666633, 1.333333333333286, 0.33333333333331555)], 1) sage: spectrum[2] # tol 1e-13 (-2.0, [(1.0, -0.2, 1.0, 0.2)], 1) sage: spectrum[3] # tol 1e-13 (-1.0, [(1.0, -0.5, 2.0, 0.5)], 1)
A generalized eigenvalue problem:
sage: A = matrix(CDF, [[1+I, -2], [3, 4]]) sage: B = matrix(CDF, [[0, 7-I], [2, -3]]) sage: E = A.eigenvectors_right(B) sage: all((A * v - e * B * v).norm() < 1e-14 for e, [v], _ in E) True
In a generalized eigenvalue problem with a singular matrix \(B\), we can check the eigenvector property using homogeneous coordinates, even though the quotient \(\alpha/\beta\) is not always defined:
sage: A = matrix.identity(RDF, 2) sage: B = matrix(RDF, [[3, 5], [6, 10]]) sage: E = A.eigenvectors_right(B, homogeneous=True) sage: all((beta * A * v - alpha * B * v).norm() < 1e-14 ....: for (alpha, beta), [v], _ in E) True
- round(ndigits=0)#
Return a copy of the matrix where all entries have been rounded to a given precision in decimal digits (default 0 digits).
INPUT:
ndigits
- The precision in number of decimal digits
OUTPUT:
A modified copy of the matrix
EXAMPLES:
sage: M = matrix(CDF, [[10.234r + 34.2343jr, 34e10r]]) sage: M [10.234 + 34.2343*I 340000000000.0] sage: M.round(2) [10.23 + 34.23*I 340000000000.0] sage: M.round() [ 10.0 + 34.0*I 340000000000.0]
- schur(base_ring=None)#
Return the Schur decomposition of the matrix.
INPUT:
base_ring
- optional, defaults to the base ring ofself
. Use this to request the base ring of the returned matrices, which will affect the format of the results.
OUTPUT:
A pair of immutable matrices. The first is a unitary matrix \(Q\). The second, \(T\), is upper-triangular when returned over the complex numbers, while it is almost upper-triangular over the reals. In the latter case, there can be some \(2\times 2\) blocks on the diagonal which represent a pair of conjugate complex eigenvalues of
self
.If
self
is the matrix \(A\), then\[A = QT({\overline Q})^t\]where the latter matrix is the conjugate-transpose of
Q
, which is also the inverse ofQ
, sinceQ
is unitary.Note that in the case of a normal matrix (Hermitian, symmetric, and others), the upper-triangular matrix is a diagonal matrix with eigenvalues of
self
on the diagonal, and the unitary matrix has columns that form an orthonormal basis composed of eigenvectors ofself
. This is known as “orthonormal diagonalization”.Warning
The Schur decomposition is not unique, as there may be numerous choices for the vectors of the orthonormal basis, and consequently different possibilities for the upper-triangular matrix. However, the diagonal of the upper-triangular matrix will always contain the eigenvalues of the matrix (in the complex version), or \(2\times 2\) block matrices in the real version representing pairs of conjugate complex eigenvalues.
In particular, results may vary across systems and processors.
EXAMPLES:
First over the complexes. The similar matrix is always upper-triangular in this case.
sage: # needs sage.symbolic sage: A = matrix(CDF, 4, 4, range(16)) + matrix(CDF, 4, 4, ....: [x^3*I for x in range(0, 16)]) sage: Q, T = A.schur() sage: (Q*Q.conjugate().transpose()).zero_at(1e-12) # tol 1e-12 [ 0.999999999999999 0.0 0.0 0.0] [ 0.0 0.9999999999999996 0.0 0.0] [ 0.0 0.0 0.9999999999999992 0.0] [ 0.0 0.0 0.0 0.9999999999999999] sage: all(T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)) True sage: (Q*T*Q.conjugate().transpose() - A).zero_at(1.0e-11) [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] sage: eigenvalues = [T[i,i] for i in range(4)]; eigenvalues [30.733... + 4648.541...*I, -0.184... - 159.057...*I, -0.523... + 11.158...*I, -0.025... - 0.642...*I] sage: A.eigenvalues() [30.733... + 4648.541...*I, -0.184... - 159.057...*I, -0.523... + 11.158...*I, -0.025... - 0.642...*I] sage: abs(A.norm()-T.norm()) < 1e-10 True
We begin with a real matrix but ask for a decomposition over the complexes. The result will yield an upper-triangular matrix over the complex numbers for
T
.sage: A = matrix(RDF, 4, 4, [x^3 for x in range(16)]) sage: Q, T = A.schur(base_ring=CDF) sage: (Q*Q.conjugate().transpose()).zero_at(1e-12) # tol 1e-12 [0.9999999999999987 0.0 0.0 0.0] [ 0.0 0.9999999999999999 0.0 0.0] [ 0.0 0.0 1.0000000000000013 0.0] [ 0.0 0.0 0.0 1.0000000000000007] sage: T.parent() Full MatrixSpace of 4 by 4 dense matrices over Complex Double Field sage: all(T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)) True sage: (Q*T*Q.conjugate().transpose() - A).zero_at(1.0e-11) [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0]
Now totally over the reals. But with complex eigenvalues, the similar matrix may not be upper-triangular. But “at worst” there may be some \(2\times 2\) blocks on the diagonal which represent a pair of conjugate complex eigenvalues. These blocks will then just interrupt the zeros below the main diagonal. This example has a pair of these of the blocks.
sage: A = matrix(RDF, 4, 4, [[1, 0, -3, -1], ....: [4, -16, -7, 0], ....: [1, 21, 1, -2], ....: [26, -1, -2, 1]]) sage: Q, T = A.schur() sage: (Q*Q.conjugate().transpose()) # tol 1e-12 [0.9999999999999994 0.0 0.0 0.0] [ 0.0 1.0000000000000013 0.0 0.0] [ 0.0 0.0 1.0000000000000004 0.0] [ 0.0 0.0 0.0 1.0000000000000016] sage: all(T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i)) False sage: all(T.zero_at(1.0e-12)[i,j] == 0 for i in range(4) for j in range(i-1)) True sage: (Q*T*Q.conjugate().transpose() - A).zero_at(1.0e-11) [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] sage: sorted(T[0:2,0:2].eigenvalues() + T[2:4,2:4].eigenvalues()) [-5.710... - 8.382...*I, -5.710... + 8.382...*I, -0.789... - 2.336...*I, -0.789... + 2.336...*I] sage: sorted(A.eigenvalues()) [-5.710... - 8.382...*I, -5.710... + 8.382...*I, -0.789... - 2.336...*I, -0.789... + 2.336...*I] sage: abs(A.norm()-T.norm()) < 1e-12 True
Starting with complex numbers and requesting a result over the reals will never happen.
sage: # needs sage.symbolic sage: A = matrix(CDF, 2, 2, [[2+I, -1+3*I], [5-4*I, 2-7*I]]) sage: A.schur(base_ring=RDF) Traceback (most recent call last): ... TypeError: unable to convert input matrix over CDF to a matrix over RDF
If theory predicts your matrix is real, but it contains some very small imaginary parts, you can specify the cutoff for “small” imaginary parts, then request the output as real matrices, and let the routine do the rest.
sage: A = matrix(RDF, 2, 2, [1, 1, -1, 0]) + matrix(CDF, 2, 2, [1.0e-14*I]*4) sage: B = A.zero_at(1.0e-12) sage: B.parent() Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field sage: Q, T = B.schur(RDF) sage: Q.parent() Full MatrixSpace of 2 by 2 dense matrices over Real Double Field sage: T.parent() Full MatrixSpace of 2 by 2 dense matrices over Real Double Field sage: Q.round(6) [ 0.707107 0.707107] [-0.707107 0.707107] sage: T.round(6) [ 0.5 1.5] [-0.5 0.5] sage: (Q*T*Q.conjugate().transpose() - B).zero_at(1.0e-11) [0.0 0.0] [0.0 0.0]
A Hermitian matrix has real eigenvalues, so the similar matrix will be upper-triangular. Furthermore, a Hermitian matrix is diagonalizable with respect to an orthonormal basis, composed of eigenvectors of the matrix. Here that basis is the set of columns of the unitary matrix.
sage: # needs sage.symbolic sage: A = matrix(CDF, [[ 52, -9*I - 8, 6*I - 187, -188*I + 2], ....: [ 9*I - 8, 12, -58*I + 59, 30*I + 42], ....: [-6*I - 187, 58*I + 59, 2677, 2264*I + 65], ....: [ 188*I + 2, -30*I + 42, -2264*I + 65, 2080]]) sage: Q, T = A.schur() sage: T = T.zero_at(1.0e-12).change_ring(RDF) sage: T.round(6) [4680.13301 0.0 0.0 0.0] [ 0.0 102.715967 0.0 0.0] [ 0.0 0.0 35.039344 0.0] [ 0.0 0.0 0.0 3.11168] sage: (Q*Q.conjugate().transpose()).zero_at(1e-12) # tol 1e-12 [1.0000000000000004 0.0 0.0 0.0] [ 0.0 0.9999999999999989 0.0 0.0] [ 0.0 0.0 1.0000000000000002 0.0] [ 0.0 0.0 0.0 0.9999999999999992] sage: (Q*T*Q.conjugate().transpose() - A).zero_at(1.0e-11) [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0]
Similarly, a real symmetric matrix has only real eigenvalues, and there is an orthonormal basis composed of eigenvectors of the matrix.
sage: A = matrix(RDF, [[ 1, -2, 5, -3], ....: [-2, 9, 1, 5], ....: [ 5, 1, 3 , 7], ....: [-3, 5, 7, -8]]) sage: Q, T = A.schur() sage: Q.round(4) [-0.3027 -0.751 0.576 -0.1121] [ 0.139 -0.3892 -0.2648 0.8713] [ 0.4361 0.359 0.7599 0.3217] [ -0.836 0.3945 0.1438 0.3533] sage: T = T.zero_at(10^-12) sage: all(abs(e) < 10^-4 ....: for e in (T - diagonal_matrix(RDF, [-13.5698, -0.8508, 7.7664, 11.6542])).list()) True sage: (Q*Q.transpose()) # tol 1e-12 [0.9999999999999998 0.0 0.0 0.0] [ 0.0 1.0 0.0 0.0] [ 0.0 0.0 0.9999999999999998 0.0] [ 0.0 0.0 0.0 0.9999999999999996] sage: (Q*T*Q.transpose() - A).zero_at(1.0e-11) [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0] [0.0 0.0 0.0 0.0]
The results are cached, both as a real factorization and also as a complex factorization. This means the returned matrices are immutable.
sage: A = matrix(RDF, 2, 2, [[0, -1], [1, 0]]) sage: Qr, Tr = A.schur(base_ring=RDF) sage: Qc, Tc = A.schur(base_ring=CDF) sage: all(M.is_immutable() for M in [Qr, Tr, Qc, Tc]) True sage: Tr.round(6) != Tc.round(6) True
AUTHOR:
Rob Beezer (2011-03-31)
- singular_values(eps=None)#
Return a sorted list of the singular values of the matrix.
INPUT:
eps
- default:None
- the largest number which will be considered to be zero. May also be set to the string ‘auto’. See the discussion below.
OUTPUT:
A sorted list of the singular values of the matrix, which are the diagonal entries of the “S” matrix in the SVD decomposition. As such, the values are real and are returned as elements of
RDF
. The list is sorted with larger values first, and since theory predicts these values are always positive, for a rank-deficient matrix the list should end in zeros (but in practice may not). The length of the list is the minimum of the row count and column count for the matrix.The number of non-zero singular values will be the rank of the matrix. However, as a numerical matrix, it is impossible to control the difference between zero entries and very small non-zero entries. As an informed consumer it is up to you to use the output responsibly. We will do our best, and give you the tools to work with the output, but we cannot give you a guarantee.
With
eps
set toNone
you will get the raw singular values and can manage them as you see fit. You may also seteps
to any positive floating point value you wish. If you seteps
to ‘auto’ this routine will compute a reasonable cutoff value, based on the size of the matrix, the largest singular value and the smallest nonzero value representable by the 53-bit precision values used. See the discussion at page 268 of [Wat2010].See the examples for a way to use the “verbose” facility to easily watch the zero cutoffs in action.
ALGORITHM:
The singular values come from the SVD decomposition computed by SciPy/NumPy.
EXAMPLES:
Singular values close to zero have trailing digits that may vary on different hardware. For exact matrices, the number of non-zero singular values will equal the rank of the matrix. So for some of the doctests we round the small singular values that ideally would be zero, to control the variability across hardware.
This matrix has a determinant of one. A chain of two or three theorems implies the product of the singular values must also be one.
sage: A = matrix(QQ, [[ 1, 0, 0, 0, 0, 1, 3], ....: [-2, 1, 1, -2, 0, -4, 0], ....: [ 1, 0, 1, -4, -6, -3, 7], ....: [-2, 2, 1, 1, 7, 1, -1], ....: [-1, 0, -1, 5, 8, 4, -6], ....: [ 4, -2, -2, 1, -3, 0, 8], ....: [-2, 1, 0, 2, 7, 3, -4]]) sage: A.determinant() 1 sage: B = A.change_ring(RDF) sage: sv = B.singular_values(); sv # tol 1e-12 [20.523980658874265, 8.486837028536643, 5.86168134845073, 2.4429165899286978, 0.5831970144724045, 0.26933287286576313, 0.0025524488076110402] sage: prod(sv) # tol 1e-12 0.9999999999999525
An exact matrix that is obviously not of full rank, and then a computation of the singular values after conversion to an approximate matrix.
sage: A = matrix(QQ, [[1/3, 2/3, 11/3], ....: [2/3, 1/3, 7/3], ....: [2/3, 5/3, 27/3]]) sage: A.rank() 2 sage: B = A.change_ring(CDF) sage: sv = B.singular_values() sage: sv[0:2] [10.1973039..., 0.487045871...] sage: sv[2] < 1e-14 True
A matrix of rank 3 over the complex numbers.
sage: A = matrix(CDF, [[46*I - 28, -47*I - 50, 21*I + 51, -62*I - 782, 13*I + 22], ....: [35*I - 20, -32*I - 46, 18*I + 43, -57*I - 670, 7*I + 3], ....: [22*I - 13, -23*I - 23, 9*I + 24, -26*I - 347, 7*I + 13], ....: [-44*I + 23, 41*I + 57, -19*I - 54, 60*I + 757, -11*I - 9], ....: [30*I - 18, -30*I - 34, 14*I + 34, -42*I - 522, 8*I + 12]]) sage: sv = A.singular_values() sage: sv[0:3] # tol 1e-14 [1440.7336659952966, 18.404403413369227, 6.839707797136151] sage: (sv[3] < 10^-13) or sv[3] True sage: (sv[4] < 10^-14) or sv[4] True
A full-rank matrix that is ill-conditioned. We use this to illustrate ways of using the various possibilities for
eps
, including one that is ill-advised. Notice that the automatically computed cutoff gets this (difficult) example slightly wrong. This illustrates the impossibility of any automated process always getting this right. Use with caution and judgement.sage: entries = [1/(i+j+1) for i in range(12) for j in range(12)] sage: B = matrix(QQ, 12, 12, entries) sage: B.rank() 12 sage: A = B.change_ring(RDF) sage: A.condition() > 1.59e16 or A.condition() True sage: A.singular_values(eps=None) # abs tol 7e-16 [1.7953720595619975, 0.38027524595503703, 0.04473854875218107, 0.0037223122378911614, 0.0002330890890217751, 1.116335748323284e-05, 4.082376110397296e-07, 1.1228610675717613e-08, 2.2519645713496478e-10, 3.1113486853814003e-12, 2.6500422260778388e-14, 9.87312834948426e-17] sage: A.singular_values(eps='auto') # abs tol 7e-16 [1.7953720595619975, 0.38027524595503703, 0.04473854875218107, 0.0037223122378911614, 0.0002330890890217751, 1.116335748323284e-05, 4.082376110397296e-07, 1.1228610675717613e-08, 2.2519645713496478e-10, 3.1113486853814003e-12, 2.6500422260778388e-14, 0.0] sage: A.singular_values(eps=1e-4) # abs tol 7e-16 [1.7953720595619975, 0.38027524595503703, 0.04473854875218107, 0.0037223122378911614, 0.0002330890890217751, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
With Sage’s “verbose” facility, you can compactly see the cutoff at work. In any application of this routine, or those that build upon it, it would be a good idea to conduct this exercise on samples. We also test here that all the values are returned in \(RDF\) since singular values are always real.
sage: A = matrix(CDF, 4, range(16)) sage: from sage.misc.verbose import set_verbose sage: set_verbose(1) sage: sv = A.singular_values(eps='auto'); sv verbose 1 (<module>) singular values, smallest-non-zero:cutoff:largest-zero, 2.2766...:6.2421...e-14:... [35.13996365902..., 2.27661020871472..., 0.0, 0.0] sage: set_verbose(0) sage: all(s in RDF for s in sv) True
AUTHOR:
Rob Beezer - (2011-02-18)
- zero_at(eps)#
Return a copy of the matrix where elements smaller than or equal to
eps
are replaced with zeroes. For complex matrices, the real and imaginary parts are considered individually.This is useful for modifying output from algorithms which have large relative errors when producing zero elements, e.g. to create reliable doctests.
INPUT:
eps
- Cutoff value
OUTPUT:
A modified copy of the matrix.
EXAMPLES:
sage: # needs sage.symbolic sage: a = matrix(CDF, [[1, 1e-4r, 1+1e-100jr], [1e-8+3j, 0, 1e-58r]]) sage: a [ 1.0 0.0001 1.0 + 1e-100*I] [ 1e-08 + 3.0*I 0.0 1e-58] sage: a.zero_at(1e-50) [ 1.0 0.0001 1.0] [1e-08 + 3.0*I 0.0 0.0] sage: a.zero_at(1e-4) [ 1.0 0.0 1.0] [3.0*I 0.0 0.0]