# Dense matrices over GF(2) using the M4RI library.¶

AUTHOR: Martin Albrecht <malb@informatik.uni-bremen.de>

EXAMPLES:

sage: a = matrix(GF(2),3,range(9),sparse=False); a
[0 1 0]
[1 0 1]
[0 1 0]
sage: a.rank()
2
sage: type(a)
<type 'sage.matrix.matrix_mod2_dense.Matrix_mod2_dense'>
sage: a[0,0] = 1
sage: a.rank()
3
sage: parent(a)
Full MatrixSpace of 3 by 3 dense matrices over Finite Field of size 2

sage: a^2
[0 1 1]
[1 0 0]
[1 0 1]
sage: a+a
[0 0 0]
[0 0 0]
[0 0 0]

sage: b = a.new_matrix(2,3,range(6)); b
[0 1 0]
[1 0 1]

sage: a*b
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Full MatrixSpace of 3 by 3 dense matrices over Finite Field of size 2' and 'Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 2'
sage: b*a
[1 0 1]
[1 0 0]

sage: TestSuite(a).run()
sage: TestSuite(b).run()

sage: a.echelonize(); a
[1 0 0]
[0 1 0]
[0 0 1]
sage: b.echelonize(); b
[1 0 1]
[0 1 0]


Todo

• make LinBox frontend and use it

• charpoly ?
• minpoly ?
• make Matrix_modn_frontend and use it (?)

class sage.matrix.matrix_mod2_dense.Matrix_mod2_dense

Dense matrix over GF(2).

augment(right, subdivide=False)

Augments self with right.

EXAMPLES:

sage: MS = MatrixSpace(GF(2),3,3)
sage: A = MS([0, 1, 0, 1, 1, 0, 1, 1, 1]); A
[0 1 0]
[1 1 0]
[1 1 1]
sage: B = A.augment(MS(1)); B
[0 1 0 1 0 0]
[1 1 0 0 1 0]
[1 1 1 0 0 1]
sage: B.echelonize(); B
[1 0 0 1 1 0]
[0 1 0 1 0 0]
[0 0 1 0 1 1]
sage: C = B.matrix_from_columns([3,4,5]); C
[1 1 0]
[1 0 0]
[0 1 1]
sage: C == ~A
True
sage: C*A == MS(1)
True


A vector may be augmented to a matrix.

sage: A = matrix(GF(2), 3, 4, range(12))
sage: v = vector(GF(2), 3, range(3))
sage: A.augment(v)
[0 1 0 1 0]
[0 1 0 1 1]
[0 1 0 1 0]


The subdivide option will add a natural subdivision between self and right. For more details about how subdivisions are managed when augmenting, see sage.matrix.matrix1.Matrix.augment().

sage: A = matrix(GF(2), 3, 5, range(15))
sage: B = matrix(GF(2), 3, 3, range(9))
sage: A.augment(B, subdivide=True)
[0 1 0 1 0|0 1 0]
[1 0 1 0 1|1 0 1]
[0 1 0 1 0|0 1 0]

density(approx=False)

Return the density of this matrix.

By density we understand the ratio of the number of nonzero positions and the self.nrows() * self.ncols(), i.e. the number of possible nonzero positions.

INPUT:

• approx – return floating point approximation (default: False)

EXAMPLES:

sage: A = random_matrix(GF(2),1000,1000)
sage: d = A.density(); d
62483/125000

sage: float(d)
0.499864

sage: A.density(approx=True)
0.499864000...

sage: float(len(A.nonzero_positions())/1000^2)
0.499864

determinant()

Return the determinant of this matrix over GF(2).

EXAMPLES:

sage: matrix(GF(2),2,[1,1,0,1]).determinant()
1
sage: matrix(GF(2),2,[1,1,1,1]).determinant()
0

echelonize(algorithm='heuristic', cutoff=0, reduced=True, **kwds)

Puts self in (reduced) row echelon form.

INPUT:

• self – a mutable matrix

• algorithm

• ‘heuristic’ – uses M4RI and PLUQ (default)
• ‘m4ri’ – uses M4RI
• ‘pluq’ – uses PLUQ factorization
• ‘classical’ – uses classical Gaussian elimination
• k – the parameter ‘k’ of the M4RI algorithm. It MUST be between 1 and 16 (inclusive). If it is not specified it will be calculated as 3/4 * log_2( min(nrows, ncols) ) as suggested in the M4RI paper.

• reduced – return reduced row echelon form (default:True)

EXAMPLES:

sage: A = random_matrix(GF(2), 10, 10)
sage: B = A.__copy__(); B.echelonize() # fastest
sage: C = A.__copy__(); C.echelonize(k=2) # force k
sage: E = A.__copy__(); E.echelonize(algorithm='classical') # force Gaussian elimination
sage: B == C == E
True


ALGORITHM:

Uses M4RI library

REFERENCES:

randomize(density=1, nonzero=False)

Randomize density proportion of the entries of this matrix, leaving the rest unchanged.

INPUT:

• density - float; proportion (roughly) to be considered for changes
• nonzero - Bool (default: False); whether the new entries are forced to be non-zero

OUTPUT:

• None, the matrix is modified in-space

EXAMPLES:

sage: A = matrix(GF(2), 5, 5, 0)
sage: A.randomize(0.5); A
[0 0 0 1 1]
[0 1 0 0 1]
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 0 1 0]
sage: A.randomize(); A
[0 0 1 1 0]
[1 1 0 0 1]
[1 1 1 1 0]
[1 1 1 1 1]
[0 0 1 1 0]

rank(algorithm='ple')

Return the rank of this matrix.

On average ‘ple’ should be faster than ‘m4ri’ and hence it is the default choice. However, for small - i.e. quite few thousand rows & columns - and sparse matrices ‘m4ri’ might be a better choice.

INPUT:

• algorithm - either “ple” or “m4ri”

EXAMPLES:

sage: A = random_matrix(GF(2), 1000, 1000)
sage: A.rank()
999

sage: A = matrix(GF(2),10, 0)
sage: A.rank()
0

row(i, from_list=False)

Return the i’th row of this matrix as a vector.

This row is a dense vector if and only if the matrix is a dense matrix.

INPUT:

• i - integer
• from_list - bool (default: False); if True, returns the i’th element of self.rows() (see rows()), which may be faster, but requires building a list of all rows the first time it is called after an entry of the matrix is changed.

EXAMPLES:

sage: A = random_matrix(GF(2),10,10); A
[0 1 0 1 1 0 0 0 1 1]
[0 1 1 1 0 1 1 0 0 1]
[0 0 0 1 0 1 0 0 1 0]
[0 1 1 0 0 1 0 1 1 0]
[0 0 0 1 1 1 1 0 1 1]
[0 0 1 1 1 1 0 0 0 0]
[1 1 1 1 0 1 0 1 1 0]
[0 0 0 1 1 0 0 0 1 1]
[1 0 0 0 1 1 1 0 1 1]
[1 0 0 1 1 0 1 0 0 0]

sage: A.row(0)
(0, 1, 0, 1, 1, 0, 0, 0, 1, 1)

sage: A.row(-1)
(1, 0, 0, 1, 1, 0, 1, 0, 0, 0)

sage: A.row(2,from_list=True)
(0, 0, 0, 1, 0, 1, 0, 0, 1, 0)

sage: A = Matrix(GF(2),1,0)
sage: A.row(0)
()

str(rep_mapping=None, zero=None, plus_one=None, minus_one=None, unicode=False, shape=None)

Return a nice string representation of the matrix.

INPUT:

• rep_mapping – a dictionary or callable used to override the usual representation of elements. For a dictionary, keys should be elements of the base ring and values the desired string representation.
• zero – string (default: None); if not None use the value of zero as the representation of the zero element.
• plus_one – string (default: None); if not None use the value of plus_one as the representation of the one element.
• minus_one – Ignored. Only for compatibility with generic matrices.
• unicode – boolean (default: False). Whether to use Unicode symbols instead of ASCII symbols for brackets and subdivision lines.
• shape – one of "square" or "round" (default: None). Switches between round and square brackets. The default depends on the setting of the unicode keyword argument. For Unicode symbols, the default is round brackets in accordance with the TeX rendering, while the ASCII rendering defaults to square brackets.

EXAMPLES:

sage: B = random_matrix(GF(2),3,3)
sage: B # indirect doctest
[0 1 0]
[0 1 1]
[0 0 0]
sage: block_matrix([[B, 1], [0, B]])
[0 1 0|1 0 0]
[0 1 1|0 1 0]
[0 0 0|0 0 1]
[-----+-----]
[0 0 0|0 1 0]
[0 0 0|0 1 1]
[0 0 0|0 0 0]
sage: B.str(zero='.')
'[. 1 .]\n[. 1 1]\n[. . .]'

sage: M = matrix.identity(GF(2), 3)
sage: M.subdivide(None, 2)
sage: print(M.str(unicode=True, shape='square'))
⎡1 0│0⎤
⎢0 1│0⎥
⎣0 0│1⎦
sage: print(unicode_art(M))  # indirect doctest
⎛1 0│0⎞
⎜0 1│0⎟
⎝0 0│1⎠

submatrix(row=0, col=0, nrows=-1, ncols=-1)

Return submatrix from the index row, col (inclusive) with dimension nrows x ncols.

INPUT:

• row – index of start row
• col – index of start column
• nrows – number of rows of submatrix
• ncols – number of columns of submatrix

EXAMPLES:

sage: A = random_matrix(GF(2),200,200)
sage: A[0:2,0:2] == A.submatrix(0,0,2,2)
True
sage: A[0:100,0:100] == A.submatrix(0,0,100,100)
True
sage: A == A.submatrix(0,0,200,200)
True

sage: A[1:3,1:3] == A.submatrix(1,1,2,2)
True
sage: A[1:100,1:100] == A.submatrix(1,1,99,99)
True
sage: A[1:200,1:200] == A.submatrix(1,1,199,199)
True


TESTS for handling of default arguments (trac ticket #18761):

sage: A.submatrix(17,15) == A.submatrix(17,15,183,185)
True
sage: A.submatrix(row=100,col=37,nrows=1,ncols=3) == A.submatrix(100,37,1,3)
True

transpose()

Returns transpose of self and leaves self untouched.

EXAMPLES:

sage: A = Matrix(GF(2),3,5,[1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0])
sage: A
[1 0 1 0 0]
[0 1 1 0 0]
[1 1 0 1 0]
sage: B = A.transpose(); B
[1 0 1]
[0 1 1]
[1 1 0]
[0 0 1]
[0 0 0]
sage: B.transpose() == A
True


.T is a convenient shortcut for the transpose:

sage: A.T
[1 0 1]
[0 1 1]
[1 1 0]
[0 0 1]
[0 0 0]

sage.matrix.matrix_mod2_dense.free_m4ri()

Free global Gray code tables.

sage.matrix.matrix_mod2_dense.from_png(filename)

Returns a dense matrix over GF(2) from a 1-bit PNG image read from filename. No attempt is made to verify that the filename string actually points to a PNG image.

INPUT:

• filename – a string

EXAMPLES:

sage: from sage.matrix.matrix_mod2_dense import from_png, to_png
sage: A = random_matrix(GF(2),10,10)
sage: fn = tmp_filename()
sage: to_png(A, fn)
sage: B = from_png(fn)
sage: A == B
True

sage.matrix.matrix_mod2_dense.parity(a)

Returns the parity of the number of bits in a.

EXAMPLES:

sage: from sage.matrix.matrix_mod2_dense import parity
sage: parity(1)
1L
sage: parity(3)
0L
sage: parity(0x10000101011)
1L

sage.matrix.matrix_mod2_dense.ple(A, algorithm='standard', param=0)

Return PLE factorization of A.

INPUT:

• A – matrix
• algorithm
• ‘standard’ asymptotically fast (default)
• ‘russian’ M4RI inspired
• ‘naive’ naive cubic
• param – either k for ‘mmpf’ is chosen or matrix multiplication cutoff for ‘standard’ (default: 0)

EXAMPLES:

sage: from sage.matrix.matrix_mod2_dense import ple
sage: A = random_matrix(GF(2),4,4); A
[0 1 0 1]
[0 1 1 1]
[0 0 0 1]
[0 1 1 0]

sage: LU, P, Q = ple(A)
sage: LU
[1 0 0 1]
[1 1 0 0]
[0 0 1 0]
[1 1 1 0]

sage: P
[0, 1, 2, 3]

sage: Q
[1, 2, 3, 3]

sage: A = random_matrix(GF(2),1000,1000)
sage: ple(A) == ple(A,'russian') == ple(A,'naive')
True

sage.matrix.matrix_mod2_dense.pluq(A, algorithm='standard', param=0)

Return PLUQ factorization of A.

INPUT:

• A – matrix
• algorithm
• ‘standard’ asymptotically fast (default)
• ‘mmpf’ M4RI inspired
• ‘naive’ naive cubic
• param – either k for ‘mmpf’ is chosen or matrix multiplication cutoff for ‘standard’ (default: 0)

EXAMPLES:

sage: from sage.matrix.matrix_mod2_dense import pluq
sage: A = random_matrix(GF(2),4,4); A
[0 1 0 1]
[0 1 1 1]
[0 0 0 1]
[0 1 1 0]

sage: LU, P, Q = pluq(A)
sage: LU
[1 0 1 0]
[1 1 0 0]
[0 0 1 0]
[1 1 1 0]

sage: P
[0, 1, 2, 3]

sage: Q
[1, 2, 3, 3]

sage.matrix.matrix_mod2_dense.to_png(A, filename)

Saves the matrix A to filename as a 1-bit PNG image.

INPUT:

• A - a matrix over GF(2)
• filename - a string for a file in a writable position

EXAMPLES:

sage: from sage.matrix.matrix_mod2_dense import from_png, to_png
sage: A = random_matrix(GF(2),10,10)
sage: fn = tmp_filename()
sage: to_png(A, fn)
sage: B = from_png(fn)
sage: A == B
True

sage.matrix.matrix_mod2_dense.unpickle_matrix_mod2_dense_v2(r, c, data, size, immutable=False)

Deserialize a matrix encoded in the string s.

INPUT:

• r – number of rows of matrix
• c – number of columns of matrix
• s – a string
• size – length of the string s
• immutable – (default: False) whether the matrix is immutable or not

EXAMPLES:

sage: A = random_matrix(GF(2),100,101)
sage: _, (r,c,s,s2,i) = A.__reduce__()
sage: from sage.matrix.matrix_mod2_dense import unpickle_matrix_mod2_dense_v2
sage: unpickle_matrix_mod2_dense_v2(r,c,s,s2,i) == A
True