Symbolic dense matrices

EXAMPLES:

sage: matrix(SR, 2, 2, range(4))
[0 1]
[2 3]
sage: matrix(SR, 2, 2, var('t'))
[t 0]
[0 t]

Arithmetic:

sage: -matrix(SR, 2, range(4))
[ 0 -1]
[-2 -3]
sage: m = matrix(SR, 2, [1..4]); sqrt(2)*m
[  sqrt(2) 2*sqrt(2)]
[3*sqrt(2) 4*sqrt(2)]
sage: m = matrix(SR, 4, [1..4^2])
sage: m * m
[ 90 100 110 120]
[202 228 254 280]
[314 356 398 440]
[426 484 542 600]

sage: m = matrix(SR, 3, [1, 2, 3]); m
[1]
[2]
[3]
sage: m.transpose() * m
[14]

Computing inverses:

sage: M = matrix(SR, 2, var('a,b,c,d'))
sage: ~M
[1/a - b*c/(a^2*(b*c/a - d))           b/(a*(b*c/a - d))]
[          c/(a*(b*c/a - d))              -1/(b*c/a - d)]
sage: (~M*M).simplify_rational()
[1 0]
[0 1]
sage: M = matrix(SR, 3, 3, range(9)) - var('t')
sage: (~M * M).simplify_rational()
[1 0 0]
[0 1 0]
[0 0 1]

sage: matrix(SR, 1, 1, 1).inverse()
[1]
sage: matrix(SR, 0, 0).inverse()
[]
sage: matrix(SR, 3, 0).inverse()
Traceback (most recent call last):
...
ArithmeticError: self must be a square matrix

Transposition:

sage: m = matrix(SR, 2, [sqrt(2), -1, pi, e^2])
sage: m.transpose()
[sqrt(2)      pi]
[     -1     e^2]

.T is a convenient shortcut for the transpose:

sage: m.T
[sqrt(2)      pi]
[     -1     e^2]

Test pickling:

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e]); m
[sqrt(2)       3]
[     pi       e]
sage: TestSuite(m).run()

Comparison:

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e])
sage: m == m
True
sage: m != 3
True
sage: m = matrix(SR,2,[1..4]); n = m^2
sage: (exp(m+n) - exp(m)*exp(n)).simplify_rational() == 0       # indirect test
True

Determinant:

sage: M = matrix(SR, 2, 2, [x,2,3,4])
sage: M.determinant()
4*x - 6
sage: M = matrix(SR, 3,3,range(9))
sage: M.det()
0
sage: t = var('t')
sage: M = matrix(SR, 2, 2, [cos(t), sin(t), -sin(t), cos(t)])
sage: M.det()
cos(t)^2 + sin(t)^2
sage: M = matrix([[sqrt(x),0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]])
sage: det(M)
sqrt(x)

Permanents:

sage: M = matrix(SR, 2, 2, [x,2,3,4])
sage: M.permanent()
4*x + 6

Rank:

sage: M = matrix(SR, 5, 5, range(25))
sage: M.rank()
2
sage: M = matrix(SR, 5, 5, range(25)) - var('t')
sage: M.rank()
5

.. warning::

    :meth:`rank` may return the wrong answer if it cannot determine that a
    matrix element that is equivalent to zero is indeed so.

Copying symbolic matrices:

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e])
sage: n = copy(m)
sage: n[0,0] = sin(1)
sage: m
[sqrt(2)       3]
[     pi       e]
sage: n
[sin(1)      3]
[    pi      e]

Conversion to Maxima:

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e])
sage: m._maxima_()
matrix([sqrt(2),3],[%pi,%e])

class sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense

Bases: Matrix_generic_dense

arguments()

Return a tuple of the arguments that self can take.

EXAMPLES:

sage: var('x,y,z')
(x, y, z)
sage: M = MatrixSpace(SR,2,2)
sage: M(x).arguments()
(x,)
sage: M(x+sin(x)).arguments()
(x,)

canonicalize_radical()

Choose a canonical branch of each entry of self by calling Expression.canonicalize_radical() componentwise.

EXAMPLES:

sage: var('x','y')
(x, y)
sage: l1 = [sqrt(2)*sqrt(3)*sqrt(6) , log(x*y)]
sage: l2 = [sin(x/(x^2 + x)) , 1]
sage: m = matrix([l1, l2])
sage: m
[sqrt(6)*sqrt(3)*sqrt(2)                log(x*y)]
[       sin(x/(x^2 + x))                       1]
sage: m.canonicalize_radical()
[              6 log(x) + log(y)]
[ sin(1/(x + 1))               1]

charpoly(var='x', algorithm=None)

Compute the characteristic polynomial of self, using maxima.

Note

The characteristic polynomial is defined as \(\det(xI-A)\).

INPUT:

• var – (default: ‘x’) name of variable of charpoly

EXAMPLES:

sage: M = matrix(SR, 2, 2, var('a,b,c,d'))
sage: M.charpoly('t')
t^2 + (-a - d)*t - b*c + a*d
sage: matrix(SR, 5, [1..5^2]).charpoly()
x^5 - 65*x^4 - 250*x^3

echelonize(**kwds)

Echelonize using the classical algorithm.

eigenvalues(extend=True)

Compute the eigenvalues by solving the characteristic polynomial in maxima.

The argument extend is ignored but kept for compatibility with other matrix classes.

EXAMPLES:

sage: a=matrix(SR,[[1,2],[3,4]])
sage: a.eigenvalues()
[-1/2*sqrt(33) + 5/2, 1/2*sqrt(33) + 5/2]

eigenvectors_left(other=None)

Compute the left eigenvectors of a matrix.

INPUT:

• other – a square matrix \(B\) (default: None) in a generalized eigenvalue problem; if None, an ordinary eigenvalue problem is solved (currently supported only if the base ring of self is RDF or CDF)

OUTPUT:

For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding left eigenspace, and n is the algebraic multiplicity of the eigenvalue.

EXAMPLES:

sage: A = matrix(SR,3,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: es = A.eigenvectors_left(); es
[(-3*sqrt(6) + 6, [(1, -1/5*sqrt(6) + 4/5, -2/5*sqrt(6) + 3/5)], 1),
 (3*sqrt(6) + 6, [(1, 1/5*sqrt(6) + 4/5, 2/5*sqrt(6) + 3/5)], 1),
 (0, [(1, -2, 1)], 1)]
sage: eval, [evec], mult = es[0]
sage: delta = eval*evec - evec*A
sage: abs(abs(delta)) < 1e-10
3/5*sqrt(((2*sqrt(6) - 3)*(sqrt(6) - 2) + 7*sqrt(6) - 18)^2 + ((sqrt(6) - 2)*(sqrt(6) - 4) + 6*sqrt(6) - 14)^2) < (1.00000000000000e-10)
sage: abs(abs(delta)).n() < 1e-10
True

sage: A = matrix(SR, 2, 2, var('a,b,c,d'))
sage: A.eigenvectors_left()
[(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
sage: es = A.eigenvectors_left(); es
[(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
sage: eval, [evec], mult = es[0]
sage: delta = eval*evec - evec*A
sage: delta.apply_map(lambda x: x.full_simplify())
(0, 0)

This routine calls Maxima and can struggle with even small matrices with a few variables, such as a \(3\times 3\) matrix with three variables. However, if the entries are integers or rationals it can produce exact values in a reasonable time. These examples create 0-1 matrices from the adjacency matrices of graphs and illustrate how the format and type of the results differ when the base ring changes. First for matrices over the rational numbers, then the same matrix but viewed as a symbolic matrix.

sage: G=graphs.CycleGraph(5)
sage: am = G.adjacency_matrix()
sage: spectrum = am.eigenvectors_left()
sage: qqbar_evalue = spectrum[2][0]
sage: type(qqbar_evalue)
<class 'sage.rings.qqbar.AlgebraicNumber'>
sage: qqbar_evalue
0.618033988749895?

sage: am = G.adjacency_matrix().change_ring(SR)
sage: spectrum = am.eigenvectors_left()
sage: symbolic_evalue = spectrum[2][0]
sage: type(symbolic_evalue)
<class 'sage.symbolic.expression.Expression'>
sage: symbolic_evalue
1/2*sqrt(5) - 1/2

sage: bool(qqbar_evalue == symbolic_evalue)
True

A slightly larger matrix with a “nice” spectrum.

sage: G = graphs.CycleGraph(6)
sage: am = G.adjacency_matrix().change_ring(SR)
sage: am.eigenvectors_left()
[(-1, [(1, 0, -1, 1, 0, -1), (0, 1, -1, 0, 1, -1)], 2), (1, [(1, 0, -1, -1, 0, 1), (0, 1, 1, 0, -1, -1)], 2), (-2, [(1, -1, 1, -1, 1, -1)], 1), (2, [(1, 1, 1, 1, 1, 1)], 1)]

eigenvectors_right(other=None)

Compute the right eigenvectors of a matrix.

INPUT:

• other – a square matrix \(B\) (default: None) in a generalized eigenvalue problem; if None, an ordinary eigenvalue problem is solved (currently supported only if the base ring of self is RDF or CDF)

OUTPUT:

For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding right eigenspace, and n is the algebraic multiplicity of the eigenvalue.

EXAMPLES:

sage: A = matrix(SR,2,2,range(4)); A
[0 1]
[2 3]
sage: right = A.eigenvectors_right(); right
[(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17) + 3/2)], 1)]

The right eigenvectors are nothing but the left eigenvectors of the transpose matrix:

sage: left  = A.transpose().eigenvectors_left(); left
[(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17) + 3/2)], 1)]
sage: right[0][1] == left[0][1]
True

exp()

Return the matrix exponential of this matrix \(X\), which is the matrix

\[e^X = \sum_{k=0}^{\infty} \frac{X^k}{k!}.\]

This function depends on maxima’s matrix exponentiation function, which does not deal well with floating point numbers. If the matrix has floating point numbers, they will be rounded automatically to rational numbers during the computation.

EXAMPLES:

sage: m = matrix(SR,2, [0,x,x,0]); m
[0 x]
[x 0]
sage: m.exp()
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]
sage: exp(m)
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]

Exponentiation works on 0x0 and 1x1 matrices, but the 1x1 example requires a patched version of maxima (github issue #32898) for now:

sage: m = matrix(SR,0,[]); m
[]
sage: m.exp()
[]
sage: m = matrix(SR,1,[2]); m
[2]
sage: m.exp()  # not tested, requires patched maxima
[e^2]

Commuting matrices \(m, n\) have the property that \(e^{m+n} = e^m e^n\) (but non-commuting matrices need not):

sage: m = matrix(SR,2,[1..4]); n = m^2
sage: m*n
[ 37  54]
[ 81 118]
sage: n*m
[ 37  54]
[ 81 118]

sage: a = exp(m+n) - exp(m)*exp(n)
sage: a.simplify_rational() == 0
True

The input matrix must be square:

sage: m = matrix(SR,2,3,[1..6]); exp(m)
Traceback (most recent call last):
...
ValueError: exp only defined on square matrices

In this example we take the symbolic answer and make it numerical at the end:

sage: exp(matrix(SR, [[1.2, 5.6], [3,4]])).change_ring(RDF)  # rel tol 1e-15
[ 346.5574872980695  661.7345909344504]
[354.50067371488416  677.4247827652946]

Another example involving the reversed identity matrix, which we clumsily create:

sage: m = identity_matrix(SR,4); m = matrix(list(reversed(m.rows()))) * x
sage: exp(m)
[1/2*(e^(2*x) + 1)*e^(-x)                        0                        0 1/2*(e^(2*x) - 1)*e^(-x)]
[                       0 1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)                        0]
[                       0 1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)                        0]
[1/2*(e^(2*x) - 1)*e^(-x)                        0                        0 1/2*(e^(2*x) + 1)*e^(-x)]

expand()

Operate point-wise on each element.

EXAMPLES:

sage: M = matrix(2, 2, range(4)) - var('x')
sage: M*M
[      x^2 + 2      -2*x + 3]
[     -4*x + 6 (x - 3)^2 + 2]
sage: (M*M).expand()
[       x^2 + 2       -2*x + 3]
[      -4*x + 6 x^2 - 6*x + 11]

factor()

Operate point-wise on each element.

EXAMPLES:

sage: M = matrix(SR, 2, 2, x^2 - 2*x + 1); M
[x^2 - 2*x + 1             0]
[            0 x^2 - 2*x + 1]
sage: M.factor()
[(x - 1)^2         0]
[        0 (x - 1)^2]

fcp(var='x')

Return the factorization of the characteristic polynomial of self.

INPUT:

• var – (default: ‘x’) name of variable of charpoly

EXAMPLES:

sage: a = matrix(SR,[[1,2],[3,4]])
sage: a.fcp()
x^2 - 5*x - 2
sage: [i for i in a.fcp()]
[(x^2 - 5*x - 2, 1)]
sage: a = matrix(SR,[[1,0],[0,2]])
sage: a.fcp()
(x - 2) * (x - 1)
sage: [i for i in a.fcp()]
[(x - 2, 1), (x - 1, 1)]
sage: a = matrix(SR, 5, [1..5^2])
sage: a.fcp()
(x^2 - 65*x - 250) * x^3
sage: list(a.fcp())
[(x^2 - 65*x - 250, 1), (x, 3)]

function(*args)

Return a matrix over a callable symbolic expression ring.

EXAMPLES:

sage: x, y = var('x,y')
sage: v = matrix([[x,y],[x*sin(y), 0]])
sage: w = v.function([x,y]); w
[       (x, y) |--> x        (x, y) |--> y]
[(x, y) |--> x*sin(y)        (x, y) |--> 0]
sage: w.parent()
Full MatrixSpace of 2 by 2 dense matrices over Callable function ring with arguments (x, y)

jordan_form(subdivide=True, transformation=False)

Return a Jordan normal form of self.

INPUT:

• self – a square matrix

• subdivide – boolean (default: True)

• transformation – boolean (default: False)

OUTPUT:

If transformation is False, only a Jordan normal form (unique up to the ordering of the Jordan blocks) is returned. Otherwise, a pair (J, P) is returned, where J is a Jordan normal form and P is an invertible matrix such that self equals P * J * P^(-1).

If subdivide is True, the Jordan blocks in the returned matrix J are indicated by a subdivision in the sense of subdivide().

EXAMPLES:

We start with some examples of diagonalisable matrices:

sage: a,b,c,d = var('a,b,c,d')
sage: matrix([a]).jordan_form()
[a]
sage: matrix([[a, 0], [1, d]]).jordan_form(subdivide=True)
[d|0]
[-+-]
[0|a]
sage: matrix([[a, 0], [1, d]]).jordan_form(subdivide=False)
[d 0]
[0 a]
sage: matrix([[a, x, x], [0, b, x], [0, 0, c]]).jordan_form()
[c|0|0]
[-+-+-]
[0|b|0]
[-+-+-]
[0|0|a]

In the following examples, we compute Jordan forms of some non-diagonalisable matrices:

sage: matrix([[a, a], [0, a]]).jordan_form()
[a 1]
[0 a]
sage: matrix([[a, 0, b], [0, c, 0], [0, 0, a]]).jordan_form()
[c|0 0]
[-+---]
[0|a 1]
[0|0 a]

The following examples illustrate the transformation flag. Note that symbolic expressions may need to be simplified to make consistency checks succeed:

sage: A = matrix([[x - a*c, a^2], [-c^2, x + a*c]])
sage: J, P = A.jordan_form(transformation=True)
sage: J, P
(
[x 1]  [-a*c    1]
[0 x], [-c^2    0]
)
sage: A1 = P * J * ~P; A1
[             -a*c + x (a*c - x)*a/c + a*x/c]
[                 -c^2               a*c + x]
sage: A1.simplify_rational() == A
True

sage: B = matrix([[a, b, c], [0, a, d], [0, 0, a]])
sage: J, T = B.jordan_form(transformation=True)
sage: J, T
(
[a 1 0]  [b*d   c   0]
[0 a 1]  [  0   d   0]
[0 0 a], [  0   0   1]
)
sage: (B * T).simplify_rational() == T * J
True

Finally, some examples involving square roots:

sage: matrix([[a, -b], [b, a]]).jordan_form()
[a - I*b|      0]
[-------+-------]
[      0|a + I*b]
sage: matrix([[a, b], [c, d]]).jordan_form(subdivide=False)
[1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2)                                                   0]
[                                                  0 1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2)]

minpoly(var='x')

Return the minimal polynomial of self.

EXAMPLES:

sage: M = Matrix.identity(SR, 2)
sage: M.minpoly()
x - 1

sage: t = var('t')
sage: m = matrix(2, [1, 2, 4, t])
sage: m.minimal_polynomial()
x^2 + (-t - 1)*x + t - 8

number_of_arguments()

Return the number of arguments that self can take.

EXAMPLES:

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: m = matrix([[a, (x+y)/(x+y)], [x^2, y^2+2]]); m
[      a       1]
[    x^2 y^2 + 2]
sage: m.number_of_arguments()
3

simplify()

Simplify self.

EXAMPLES:

sage: var('x,y,z')
(x, y, z)
sage: m = matrix([[z, (x+y)/(x+y)], [x^2, y^2+2]]); m
[      z       1]
[    x^2 y^2 + 2]
sage: m.simplify()
[      z       1]
[    x^2 y^2 + 2]

simplify_full()

Simplify a symbolic matrix by calling Expression.simplify_full() componentwise.

INPUT:

• self – the matrix whose entries we should simplify.

OUTPUT:

A copy of self with all of its entries simplified.

EXAMPLES:

Symbolic matrices will have their entries simplified:

sage: a,n,k = SR.var('a,n,k')
sage: f1 = sin(x)^2 + cos(x)^2
sage: f2 = sin(x/(x^2 + x))
sage: f3 = binomial(n,k)*factorial(k)*factorial(n-k)
sage: f4 = x*sin(2)/(x^a)
sage: A = matrix(SR, [[f1,f2],[f3,f4]])
sage: A.simplify_full()
[                1    sin(1/(x + 1))]
[     factorial(n) x^(-a + 1)*sin(2)]

simplify_rational()

EXAMPLES:

sage: M = matrix(SR, 3, 3, range(9)) - var('t')
sage: (~M*M)[0,0]
t*(3*(2/t + (6/t + 7)/((t - 3/t - 4)*t))*(2/t + (6/t + 5)/((t - 3/t
- 4)*t))/(t - (6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) + 1/t +
3/((t - 3/t - 4)*t^2)) - 6*(2/t + (6/t + 5)/((t - 3/t - 4)*t))/(t -
(6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) - 3*(6/t + 7)*(2/t +
(6/t + 5)/((t - 3/t - 4)*t))/((t - (6/t + 7)*(6/t + 5)/(t - 3/t -
4) - 12/t - 8)*(t - 3/t - 4)) - 3/((t - 3/t - 4)*t)
sage: expand((~M*M)[0,0])
1
sage: (~M * M).simplify_rational()
[1 0 0]
[0 1 0]
[0 0 1]

simplify_trig()

EXAMPLES:

sage: theta = var('theta')
sage: M = matrix(SR, 2, 2, [cos(theta), sin(theta), -sin(theta), cos(theta)])
sage: ~M
[1/cos(theta) - sin(theta)^2/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta)^2)                   -sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))]
[                   sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))                                          1/(sin(theta)^2/cos(theta) + cos(theta))]
sage: (~M).simplify_trig()
[ cos(theta) -sin(theta)]
[ sin(theta)  cos(theta)]

variables()

Return the variables of self.

EXAMPLES:

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: m = matrix([[x, x+2], [x^2, x^2+2]]); m
[      x   x + 2]
[    x^2 x^2 + 2]
sage: m.variables()
(x,)
sage: m = matrix([[a, b+c], [x^2, y^2+2]]); m
[      a   b + c]
[    x^2 y^2 + 2]
sage: m.variables()
(a, b, c, x, y)