Constructors for special matrices#

This module gathers several constructors for special, commonly used or interesting matrices. These can be reached through matrix.<tab>.

For example, here is a circulant matrix of order five:

sage: matrix.circulant(SR.var('a b c d e'))
[a b c d e]
[e a b c d]
[d e a b c]
[c d e a b]
[b c d e a]

The following constructions are available:

block_diagonal_matrix()

elementary_matrix()

hankel()

hilbert()

identity_matrix()

ith_to_zero_rotation_matrix()

random_diagonalizable_matrix()

random_echelonizable_matrix()

random_rref_matrix()

random_subspaces_matrix()

random_unimodular_matrix()

toeplitz()

vandermonde()

vector_on_axis_rotation_matrix()

zero_matrix()

The Combinatorics module provides further matrix constructors, such as Hadamard matrices and Latin squares. See:

sage.combinat.matrices.hadamard_matrix

sage.combinat.matrices.latin

sage.matrix.special.block_diagonal_matrix(*sub_matrices, **kwds)#

This function is available as block_diagonal_matrix(…) and matrix.block_diagonal(…).

Create a block matrix whose diagonal block entries are given by sub_matrices, with zero elsewhere.

See also block_matrix().

EXAMPLES:

sage: A = matrix(ZZ, 2, [1,2,3,4])
sage: block_diagonal_matrix(A, A)
[1 2|0 0]
[3 4|0 0]
[---+---]
[0 0|1 2]
[0 0|3 4]

The sub-matrices need not be square:

sage: B = matrix(QQ, 2, 3, range(6))
sage: block_diagonal_matrix(~A, B)
[  -2    1|   0    0    0]
[ 3/2 -1/2|   0    0    0]
[---------+--------------]
[   0    0|   0    1    2]
[   0    0|   3    4    5]

sage.matrix.special.block_matrix(*args, **kwds)#

This function is available as block_matrix(…) and matrix.block(…).

Return a larger matrix made by concatenating submatrices (rows first, then columns). For example, the matrix

[ A B ]
[ C D ]

is made up of submatrices A, B, C, and D.

INPUT:

The block_matrix command takes a list of submatrices to add as blocks, optionally preceded by a ring and the number of block rows and block columns, and returns a matrix.

The submatrices can be specified as a list of matrices (using nrows and ncols to determine their layout), or a list of lists of matrices, where each list forms a row.

ring - the base ring
nrows - the number of block rows
ncols - the number of block cols
sub_matrices - matrices (see below for syntax)
subdivide - boolean, whether or not to add subdivision information to the matrix
sparse - boolean, whether to make the resulting matrix sparse

EXAMPLES:

sage: A = matrix(QQ, 2, 2, [3,9,6,10])
sage: block_matrix([ [A, -A], [~A, 100*A] ])
[    3     9|   -3    -9]
[    6    10|   -6   -10]
[-----------+-----------]
[-5/12   3/8|  300   900]
[  1/4  -1/8|  600  1000]

If the number of submatrices in each row is the same, you can specify the submatrices as a single list too:

sage: block_matrix(2, 2, [ A, A, A, A ])
[ 3  9| 3  9]
[ 6 10| 6 10]
[-----+-----]
[ 3  9| 3  9]
[ 6 10| 6 10]

One can use constant entries:

sage: block_matrix([ [1, A], [0, 1] ])
[ 1  0| 3  9]
[ 0  1| 6 10]
[-----+-----]
[ 0  0| 1  0]
[ 0  0| 0  1]

A zero entry may represent any square or non-square zero matrix:

sage: B = matrix(QQ, 1, 1, [ 1 ] )
sage: C = matrix(QQ, 2, 2, [ 2, 3, 4, 5 ] )
sage: block_matrix([ [B, 0], [0, C] ])
[1|0 0]
[-+---]
[0|2 3]
[0|4 5]

One can specify the number of rows or columns as keywords too:

sage: block_matrix([A, -A, ~A, 100*A], ncols=4)
[    3     9|   -3    -9|-5/12   3/8|  300   900]
[    6    10|   -6   -10|  1/4  -1/8|  600  1000]

sage: block_matrix([A, -A, ~A, 100*A], nrows=1)
[    3     9|   -3    -9|-5/12   3/8|  300   900]
[    6    10|   -6   -10|  1/4  -1/8|  600  1000]

It handles base rings nicely too:

sage: R.<x> = ZZ['x']
sage: block_matrix(2, 2, [1/2, A, 0, x-1])
[  1/2     0|    3     9]
[    0   1/2|    6    10]
[-----------+-----------]
[    0     0|x - 1     0]
[    0     0|    0 x - 1]

sage: block_matrix(2, 2, [1/2, A, 0, x-1]).parent()
Full MatrixSpace of 4 by 4 dense matrices over Univariate Polynomial Ring in x over Rational Field

Subdivisions are optional. If they are disabled, the columns need not line up:

sage: B = matrix(QQ, 2, 3, range(6))
sage: block_matrix([ [~A, B], [B, ~A] ], subdivide=False)
[-5/12   3/8     0     1     2]
[  1/4  -1/8     3     4     5]
[    0     1     2 -5/12   3/8]
[    3     4     5   1/4  -1/8]

Without subdivisions it also deduces dimensions for scalars if possible:

sage: C = matrix(ZZ, 1, 2, range(2))
sage: block_matrix([ [ C, 0 ], [ 3, 4 ], [ 5, 6, C ] ], subdivide=False )
[0 1 0 0]
[3 0 4 0]
[0 3 0 4]
[5 6 0 1]

If all submatrices are sparse (unless there are none at all), the result will be a sparse matrix. Otherwise it will be dense by default. The sparse keyword can be used to override this:

sage: A = Matrix(ZZ, 2, 2, [0, 1, 0, 0], sparse=True)
sage: block_matrix([ [ A ], [ A ] ]).parent()
Full MatrixSpace of 4 by 2 sparse matrices over Integer Ring
sage: block_matrix([ [ A ], [ A ] ], sparse=False).parent()
Full MatrixSpace of 4 by 2 dense matrices over Integer Ring

Consecutive zero submatrices are consolidated.

sage: B = matrix(2, range(4))
sage: C = matrix(2, 8, range(16))
sage: block_matrix(2, [[B,0,0,B],[C]], subdivide=False)
[ 0  1  0  0  0  0  0  1]
[ 2  3  0  0  0  0  2  3]
[ 0  1  2  3  4  5  6  7]
[ 8  9 10 11 12 13 14 15]

Ambiguity is not tolerated.

sage: B = matrix(2, range(4))
sage: C = matrix(2, 8, range(16))
sage: block_matrix(2, [[B,0,B,0],[C]], subdivide=False)
Traceback (most recent call last):
...
ValueError: insufficient information to determine submatrix widths

Giving only a flat list of submatrices does not work:

sage: A = matrix(2, 3, range(6))
sage: B = matrix(3, 3, range(9))
sage: block_matrix([A, A, B, B])
Traceback (most recent call last):
...
ValueError: must specify either nrows or ncols

sage.matrix.special.circulant(v, sparse=None)#

This function is available as circulant(…) and matrix.circulant(…).

Return the circulant matrix specified by its 1st row \(v\)

A circulant \(n \times n\) matrix specified by the 1st row \(v=(v_0...v_{n-1})\) is the matrix \((c_{ij})_{0 \leq i,j\leq n-1}\), where \(c_{ij}=v_{j-i \mod b}\).

INPUT:

v – a list or a vector of values
sparse – None by default; if sparse is set to True, the output will be sparse. Respectively, setting it to False produces dense output. If sparse is not set, and if v is a vector, the output sparsity is determined by the sparsity of v; else, the output will be dense.

EXAMPLES:

sage: v=[1,2,3,4,8]
sage: matrix.circulant(v)
[1 2 3 4 8]
[8 1 2 3 4]
[4 8 1 2 3]
[3 4 8 1 2]
[2 3 4 8 1]
sage: m = matrix.circulant(vector(GF(3),[0,1,-1],sparse=True)); m
[0 1 2]
[2 0 1]
[1 2 0]
sage: m.is_sparse()
True

sage.matrix.special.column_matrix(*args, **kwds)#

This function is available as column_matrix(…) and matrix.column(…).

Construct a matrix, and then swap rows for columns and columns for rows.

Note

Linear algebra in Sage favors rows over columns. So, generally, when creating a matrix, input vectors and lists are treated as rows. This function is a convenience that turns around this convention when creating a matrix. If you are not familiar with the usual matrix() constructor, you might want to consider it first.

INPUT:

Inputs are almost exactly the same as for the matrix() constructor, which are documented there. But see examples below for how dimensions are handled.

OUTPUT:

Output is exactly the transpose of what the matrix() constructor would return. In other words, the matrix constructor builds a matrix and then this function exchanges rows for columns, and columns for rows.

EXAMPLES:

The most compelling use of this function is when you have a collection of lists or vectors that you would like to become the columns of a matrix. In almost any other situation, the matrix`() constructor can probably do the job just as easily, or easier.

sage: col_1 = [1,2,3]
sage: col_2 = [4,5,6]
sage: column_matrix([col_1, col_2])
[1 4]
[2 5]
[3 6]

sage: v1 = vector(QQ, [10, 20])
sage: v2 = vector(QQ, [30, 40])
sage: column_matrix(QQ, [v1, v2])
[10 30]
[20 40]

If you only specify one dimension along with a flat list of entries, then it will be the number of columns in the result (which is different from the behavior of the matrix constructor).

sage: column_matrix(ZZ, 8, range(24))
[ 0  3  6  9 12 15 18 21]
[ 1  4  7 10 13 16 19 22]
[ 2  5  8 11 14 17 20 23]

And when you specify two dimensions, then they should be number of columns first, then the number of rows, which is the reverse of how they would be specified for the matrix constructor.

sage: column_matrix(QQ, 5, 3, range(15))
[ 0  3  6  9 12]
[ 1  4  7 10 13]
[ 2  5  8 11 14]

And a few unproductive, but illustrative, examples.

sage: A = matrix(ZZ, 3, 4, range(12))
sage: B = column_matrix(ZZ, 3, 4, range(12))
sage: A == B.transpose()
True

sage: A = matrix(QQ, 7, 12, range(84))
sage: A == column_matrix(A.columns())
True

sage: A = column_matrix(QQ, matrix(ZZ, 3, 2, range(6)) )
sage: A
[0 2 4]
[1 3 5]
sage: A.parent()
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

sage.matrix.special.companion_matrix(poly, format='right')#

This function is available as companion_matrix(…) and matrix.companion(…).

Create a companion matrix from a monic polynomial.

INPUT:

poly – a univariate polynomial, or an iterable containing the coefficients of a polynomial, with low-degree coefficients first. The polynomial (or the polynomial implied by the coefficients) must be monic. In other words, the leading coefficient must be one. A symbolic expression that might also be a polynomial is not proper input, see examples below.
format – default: ‘right’ - specifies one of four variations of a companion matrix. Allowable values are ‘right’, ‘left’, ‘top’ and ‘bottom’, which indicates which border of the matrix contains the negatives of the coefficients.

OUTPUT:

A square matrix with a size equal to the degree of the polynomial. The returned matrix has ones above, or below the diagonal, and the negatives of the coefficients along the indicated border of the matrix (excepting the leading one coefficient). See the first examples below for precise illustrations.

EXAMPLES:

Each of the four possibilities. Notice that the coefficients are specified and their negatives become the entries of the matrix. The leading one must be given, but is not used. The permutation matrix P is the identity matrix, with the columns reversed. The last three statements test the general relationships between the four variants.

sage: poly = [-2, -3, -4, -5, -6, 1]
sage: R = companion_matrix(poly, format='right'); R
[0 0 0 0 2]
[1 0 0 0 3]
[0 1 0 0 4]
[0 0 1 0 5]
[0 0 0 1 6]
sage: L = companion_matrix(poly, format='left'); L
[6 1 0 0 0]
[5 0 1 0 0]
[4 0 0 1 0]
[3 0 0 0 1]
[2 0 0 0 0]
sage: B = companion_matrix(poly, format='bottom'); B
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[2 3 4 5 6]
sage: T = companion_matrix(poly, format='top'); T
[6 5 4 3 2]
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]

sage: perm = Permutation([5, 4, 3, 2, 1])
sage: P = perm.to_matrix()
sage: L == P*R*P
True
sage: B == R.transpose()
True
sage: T == P*R.transpose()*P
True

A polynomial may be used as input, however a symbolic expression, even if it looks like a polynomial, is not regarded as such when used as input to this routine. Obtaining the list of coefficients from a symbolic polynomial is one route to the companion matrix.

sage: x = polygen(QQ, 'x')
sage: p = x^3 - 4*x^2 + 8*x - 12
sage: companion_matrix(p)
[ 0  0 12]
[ 1  0 -8]
[ 0  1  4]

sage: y = var('y')
sage: q = y^3 -2*y + 1
sage: companion_matrix(q)
Traceback (most recent call last):
...
TypeError: input must be a polynomial (not a symbolic expression, see docstring), or other iterable, not y^3 - 2*y + 1

sage: coeff_list = [q(y=0)] + [q.coefficient(y^k) for k in range(1, q.degree(y)+1)]
sage: coeff_list
[1, -2, 0, 1]
sage: companion_matrix(coeff_list)
[ 0  0 -1]
[ 1  0  2]
[ 0  1  0]

The minimal polynomial of a companion matrix is equal to the polynomial used to create it. Used in a block diagonal construction, they can be used to create matrices with any desired minimal polynomial, or characteristic polynomial.

sage: t = polygen(QQ, 't')
sage: p = t^12 - 7*t^4 + 28*t^2 - 456
sage: C = companion_matrix(p, format='top')
sage: q = C.minpoly(var='t'); q
t^12 - 7*t^4 + 28*t^2 - 456
sage: p == q
True

sage: p = t^3 + 3*t - 8
sage: q = t^5 + t - 17
sage: A = block_diagonal_matrix( companion_matrix(p),
....:                            companion_matrix(p^2),
....:                            companion_matrix(q),
....:                            companion_matrix(q) )
sage: A.charpoly(var='t').factor()
(t^3 + 3*t - 8)^3 * (t^5 + t - 17)^2
sage: A.minpoly(var='t').factor()
(t^3 + 3*t - 8)^2 * (t^5 + t - 17)

AUTHOR:

Rob Beezer (2011-05-19)

sage.matrix.special.diagonal_matrix(arg0=None, arg1=None, arg2=None, sparse=True)#

This function is available as diagonal_matrix(…) and matrix.diagonal(…).

Return a square matrix with specified diagonal entries, and zeros elsewhere.

FORMATS:

diagonal_matrix(entries)

diagonal_matrix(nrows, entries)

diagonal_matrix(ring, entries)

diagonal_matrix(ring, nrows, entries)

INPUT:

entries - the values to place along the diagonal of the returned matrix. This may be a flat list, a flat tuple, a vector or free module element, or a one-dimensional NumPy array.
nrows - the size of the returned matrix, which will have an equal number of columns
ring - the ring containing the entries of the diagonal entries. This may not be specified in combination with a NumPy array.
sparse - default: True - whether or not the result has a sparse implementation.

OUTPUT:

A square matrix over the given ring with a size given by nrows. If the ring is not given it is inferred from the given entries. The values on the diagonal of the returned matrix come from entries. If the number of entries is not enough to fill the whole diagonal, it is padded with zeros.

EXAMPLES:

We first demonstrate each of the input formats with various different ways to specify the entries.

Format 1: a flat list of entries.

sage: A = diagonal_matrix([2, 1.3, 5]); A
[ 2.00000000000000 0.000000000000000 0.000000000000000]
[0.000000000000000  1.30000000000000 0.000000000000000]
[0.000000000000000 0.000000000000000  5.00000000000000]
sage: A.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Real Field with 53 bits of precision

Format 2: size specified, a tuple with initial entries. Note that a short list of entries is effectively padded with zeros.

sage: A = diagonal_matrix(3, (4, 5)); A
[4 0 0]
[0 5 0]
[0 0 0]
sage: A.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring

Format 3: ring specified, a vector of entries.

sage: A = diagonal_matrix(QQ, vector(ZZ, [1,2,3])); A
[1 0 0]
[0 2 0]
[0 0 3]
sage: A.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Rational Field

Format 4: ring, size and list of entries.

sage: A = diagonal_matrix(FiniteField(3), 3, [2, 16]); A
[2 0 0]
[0 1 0]
[0 0 0]
sage: A.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Finite Field of size 3

NumPy arrays may be used as input.

sage: import numpy
sage: entries = numpy.array([1.2, 5.6]); entries
array([1.2, 5.6])
sage: A = diagonal_matrix(3, entries); A
[1.2 0.0 0.0]
[0.0 5.6 0.0]
[0.0 0.0 0.0]
sage: A.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Real Double Field

sage: j = complex(0,1)
sage: entries = numpy.array([2.0+j, 8.1, 3.4+2.6*j]); entries
array([2. +1.j , 8.1+0.j , 3.4+2.6j])
sage: A = diagonal_matrix(entries); A
[2.0 + 1.0*I         0.0         0.0]
[        0.0         8.1         0.0]
[        0.0         0.0 3.4 + 2.6*I]
sage: A.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Complex Double Field

sage: entries = numpy.array([4, 5, 6])
sage: A = diagonal_matrix(entries); A
[4 0 0]
[0 5 0]
[0 0 6]
sage: A.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring

sage: entries = numpy.array([4.1, 5.2, 6.3])
sage: A = diagonal_matrix(ZZ, entries); A
Traceback (most recent call last):
...
TypeError: unable to convert 4.1 to an element of Integer Ring

By default returned matrices have a sparse implementation. This can be changed when using any of the formats.

sage: A = diagonal_matrix([1,2,3], sparse=False)
sage: A.parent()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring

An empty list and no ring specified defaults to the integers.

sage: A = diagonal_matrix([])
sage: A.parent()
Full MatrixSpace of 0 by 0 sparse matrices over Integer Ring

Giving the entries improperly may first complain about not being iterable:

sage: diagonal_matrix(QQ, 5, 10)
Traceback (most recent call last):
...
TypeError: 'sage.rings.integer.Integer' object is not iterable

Giving too many entries will raise an error.

sage: diagonal_matrix(QQ, 3, [1,2,3,4])
Traceback (most recent call last):
...
ValueError: number of diagonal matrix entries (4) exceeds the requested matrix size (3)

A negative size sometimes causes the error that there are too many elements.

sage: diagonal_matrix(-2, [2])
Traceback (most recent call last):
...
ValueError: number of diagonal matrix entries (1) exceeds the requested matrix size (-2)

Types for the entries need to be iterable (tuple, list, vector, NumPy array, etc):

sage: diagonal_matrix(x^2)
Traceback (most recent call last):
...
TypeError: 'sage.symbolic.expression.Expression' object is not iterable

AUTHOR:

Rob Beezer (2011-01-11): total rewrite

sage.matrix.special.elementary_matrix(arg0, arg1=None, **kwds)#

This function is available as elementary_matrix(…) and matrix.elementary(…).

Creates a square matrix that corresponds to a row operation or a column operation.

FORMATS:

In each case, R is the base ring, and is optional. n is the size of the square matrix created. Any call may include the sparse keyword to determine the representation used. The default is False which leads to a dense representation. We describe the matrices by their associated row operation, see the output description for more.

elementary_matrix(R, n, row1=i, row2=j)

The matrix which swaps rows i and j.
elementary_matrix(R, n, row1=i, scale=s)

The matrix which multiplies row i by s.
elementary_matrix(R, n, row1=i, row2=j, scale=s)

The matrix which multiplies row j by s and adds it to row i.

Elementary matrices representing column operations are created in an entirely analogous way, replacing row1 by col1 and replacing row2 by col2.

Specifying the ring for entries of the matrix is optional. If it is not given, and a scale parameter is provided, then a ring containing the value of scale will be used. Otherwise, the ring defaults to the integers.

OUTPUT:

An elementary matrix is a square matrix that is very close to being an identity matrix. If E is an elementary matrix and A is any matrix with the same number of rows, then E*A is the result of applying a row operation to A. This is how the three types created by this function are described. Similarly, an elementary matrix can be associated with a column operation, so if E has the same number of columns as A then A*E is the result of performing a column operation on A.

An elementary matrix representing a row operation is created if row1 is specified, while an elementary matrix representing a column operation is created if col1 is specified.

EXAMPLES:

Over the integers, creating row operations. Recall that row and column numbering begins at zero.

sage: A = matrix(ZZ, 4, 10, range(40)); A
[ 0  1  2  3  4  5  6  7  8  9]
[10 11 12 13 14 15 16 17 18 19]
[20 21 22 23 24 25 26 27 28 29]
[30 31 32 33 34 35 36 37 38 39]

sage: E = elementary_matrix(4, row1=1, row2=3); E
[1 0 0 0]
[0 0 0 1]
[0 0 1 0]
[0 1 0 0]
sage: E*A
[ 0  1  2  3  4  5  6  7  8  9]
[30 31 32 33 34 35 36 37 38 39]
[20 21 22 23 24 25 26 27 28 29]
[10 11 12 13 14 15 16 17 18 19]

sage: E = elementary_matrix(4, row1=2, scale=10); E
[ 1  0  0  0]
[ 0  1  0  0]
[ 0  0 10  0]
[ 0  0  0  1]
sage: E*A
[  0   1   2   3   4   5   6   7   8   9]
[ 10  11  12  13  14  15  16  17  18  19]
[200 210 220 230 240 250 260 270 280 290]
[ 30  31  32  33  34  35  36  37  38  39]

sage: E = elementary_matrix(4, row1=2, row2=1, scale=10); E
[ 1  0  0  0]
[ 0  1  0  0]
[ 0 10  1  0]
[ 0  0  0  1]
sage: E*A
[  0   1   2   3   4   5   6   7   8   9]
[ 10  11  12  13  14  15  16  17  18  19]
[120 131 142 153 164 175 186 197 208 219]
[ 30  31  32  33  34  35  36  37  38  39]

Over the rationals, now as column operations. Recall that row and column numbering begins at zero. Checks now have the elementary matrix on the right.

sage: A = matrix(QQ, 5, 4, range(20)); A
[ 0  1  2  3]
[ 4  5  6  7]
[ 8  9 10 11]
[12 13 14 15]
[16 17 18 19]

sage: E = elementary_matrix(QQ, 4, col1=1, col2=3); E
[1 0 0 0]
[0 0 0 1]
[0 0 1 0]
[0 1 0 0]
sage: A*E
[ 0  3  2  1]
[ 4  7  6  5]
[ 8 11 10  9]
[12 15 14 13]
[16 19 18 17]

sage: E = elementary_matrix(QQ, 4, col1=2, scale=1/2); E
[  1   0   0   0]
[  0   1   0   0]
[  0   0 1/2   0]
[  0   0   0   1]
sage: A*E
[ 0  1  1  3]
[ 4  5  3  7]
[ 8  9  5 11]
[12 13  7 15]
[16 17  9 19]

sage: E = elementary_matrix(QQ, 4, col1=2, col2=1, scale=10); E
[ 1  0  0  0]
[ 0  1 10  0]
[ 0  0  1  0]
[ 0  0  0  1]
sage: A*E
[  0   1  12   3]
[  4   5  56   7]
[  8   9 100  11]
[ 12  13 144  15]
[ 16  17 188  19]

An elementary matrix is always nonsingular. Then repeated row operations can be represented by products of elementary matrices, and this product is again nonsingular. If row operations are to preserve fundamental properties of a matrix (like rank), we do not allow scaling a row by zero. Similarly, the corresponding elementary matrix is not constructed. Also, we do not allow adding a multiple of a row to itself, since this could also lead to a new zero row.

sage: A = matrix(QQ, 4, 10, range(40)); A
[ 0  1  2  3  4  5  6  7  8  9]
[10 11 12 13 14 15 16 17 18 19]
[20 21 22 23 24 25 26 27 28 29]
[30 31 32 33 34 35 36 37 38 39]

sage: E1 = elementary_matrix(QQ, 4, row1=0, row2=1)
sage: E2 = elementary_matrix(QQ, 4, row1=3, row2=0, scale=100)
sage: E = E2*E1
sage: E.is_singular()
False
sage: E*A
[  10   11   12   13   14   15   16   17   18   19]
[   0    1    2    3    4    5    6    7    8    9]
[  20   21   22   23   24   25   26   27   28   29]
[1030 1131 1232 1333 1434 1535 1636 1737 1838 1939]

sage: E3 = elementary_matrix(QQ, 4, row1=3, scale=0)
Traceback (most recent call last):
...
ValueError: scale parameter of row of elementary matrix must be non-zero

sage: E4 = elementary_matrix(QQ, 4, row1=3, row2=3, scale=12)
Traceback (most recent call last):
...
ValueError: cannot add a multiple of a row to itself

If the ring is not specified, and a scale parameter is given, the base ring for the matrix is chosen to contain the scale parameter. Otherwise, if no ring is given, the default is the integers.

sage: E = elementary_matrix(4, row1=1, row2=3)
sage: E.parent()
Full MatrixSpace of 4 by 4 dense matrices over Integer Ring

sage: E = elementary_matrix(4, row1=1, scale=4/3)
sage: E.parent()
Full MatrixSpace of 4 by 4 dense matrices over Rational Field

sage: E = elementary_matrix(4, row1=1, scale=I)
sage: E.parent()
Full MatrixSpace of 4 by 4 dense matrices over Number Field in I with defining polynomial x^2 + 1 with I = 1*I

sage: E = elementary_matrix(4, row1=1, scale=CDF(I))
sage: E.parent()
Full MatrixSpace of 4 by 4 dense matrices over Complex Double Field

sage: E = elementary_matrix(4, row1=1, scale=QQbar(I))
sage: E.parent()
Full MatrixSpace of 4 by 4 dense matrices over Algebraic Field

Returned matrices have a dense implementation by default, but a sparse implementation may be requested.

sage: E = elementary_matrix(4, row1=0, row2=1)
sage: E.is_dense()
True

sage: E = elementary_matrix(4, row1=0, row2=1, sparse=True)
sage: E.is_sparse()
True

And the ridiculously small cases. The zero-row matrix cannot be built since then there are no rows to manipulate.

sage: elementary_matrix(QQ, 1, row1=0, row2=0)
[1]
sage: elementary_matrix(QQ, 0, row1=0, row2=0)
Traceback (most recent call last):
...
ValueError: size of elementary matrix must be 1 or greater, not 0

AUTHOR:

Rob Beezer (2011-03-04)

sage.matrix.special.hankel(c, r=None, ring=None)#

This function is available as hankel(…) and matrix.hankel(…).

Return a Hankel matrix of given first column and whose elements are zero below the first anti-diagonal.

The Hankel matrix is symmetric and constant across the anti-diagonals, with elements

\[H_{ij} = v_{i+j-1},\qquad i = 1,\ldots, m,~j = 1,\ldots, n,\]

where the vector \(v_i = c_i\) for \(i = 1,\ldots, m\) and \(v_{m+i} = r_i\) for \(i = 1, \ldots, n-1\) completely determines the Hankel matrix. If the last row, \(r\), is not given, the Hankel matrix is square by default and \(r = 0\). For more information see the Wikipedia article Hankel_matrix.

INPUT:

c – vector, first column of the Hankel matrix
r – vector (optional, default: None), last row of the Hankel matrix, from the second to the last column
ring – base ring (optional, default: None) of the resulting matrix

EXAMPLES:

A Hankel matrix with symbolic entries:

sage: matrix.hankel(SR.var('a, b, c, d, e'))
[a b c d e]
[b c d e 0]
[c d e 0 0]
[d e 0 0 0]
[e 0 0 0 0]

We can also pass the elements of the last row, starting at the second column:

sage: matrix.hankel(SR.var('a, b, c, d, e'), SR.var('f, g, h, i'))
[a b c d e]
[b c d e f]
[c d e f g]
[d e f g h]
[e f g h i]

A third order Hankel matrix in the integers:

sage: matrix.hankel([1, 2, 3])
[1 2 3]
[2 3 0]
[3 0 0]

The second argument allows to customize the last row:

sage: matrix.hankel([1..3], [7..10])
[ 1  2  3  7  8]
[ 2  3  7  8  9]
[ 3  7  8  9 10]

sage.matrix.special.hilbert(dim, ring=Rational Field)#

This function is available as hilbert(…) and matrix.hilbert(…).

Return a Hilbert matrix of the given dimension.

The \(n\) dimensional Hilbert matrix is a square matrix with entries being unit fractions,

\[H_{ij} = \frac{1}{i+j-1},\qquad i, j = 1,\ldots, n.\]

For more information see the Wikipedia article Hilbert_matrix.

INPUT:

dim – integer, the dimension of the Hilbert matrix
ring – base ring (optional, default: \QQ) of the resulting matrix

EXAMPLES:

sage: matrix.hilbert(5)
[  1 1/2 1/3 1/4 1/5]
[1/2 1/3 1/4 1/5 1/6]
[1/3 1/4 1/5 1/6 1/7]
[1/4 1/5 1/6 1/7 1/8]
[1/5 1/6 1/7 1/8 1/9]

sage.matrix.special.identity_matrix(ring, n=0, sparse=False)#

This function is available as identity_matrix(…) and matrix.identity(…).

Return the \(n \times n\) identity matrix over the given ring.

The default ring is the integers.

EXAMPLES:

sage: M = identity_matrix(QQ, 2); M
[1 0]
[0 1]
sage: M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: M = identity_matrix(2); M
[1 0]
[0 1]
sage: M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: M.is_mutable()
True
sage: M = identity_matrix(3, sparse=True); M
[1 0 0]
[0 1 0]
[0 0 1]
sage: M.parent()
Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring
sage: M.is_mutable()
True

sage.matrix.special.ith_to_zero_rotation_matrix(v, i, ring=None)#

This function is available as ith_to_zero_rotation_matrix(…) and matrix.ith_to_zero_rotation(…).

Return a rotation matrix that sends the i-th coordinates of the vector v to zero by doing a rotation with the (i-1)-th coordinate.

INPUT:

v – vector
i – integer
ring – ring (optional, default: None) of the resulting matrix

OUTPUT:

A matrix

EXAMPLES:

sage: from sage.matrix.constructor import ith_to_zero_rotation_matrix
sage: v = vector((1,2,3))
sage: ith_to_zero_rotation_matrix(v, 2)
[             1              0              0]
[             0  2/13*sqrt(13)  3/13*sqrt(13)]
[             0 -3/13*sqrt(13)  2/13*sqrt(13)]
sage: ith_to_zero_rotation_matrix(v, 2) * v
(1, sqrt(13), 0)

sage: ith_to_zero_rotation_matrix(v, 0)
[ 3/10*sqrt(10)              0 -1/10*sqrt(10)]
[             0              1              0]
[ 1/10*sqrt(10)              0  3/10*sqrt(10)]
sage: ith_to_zero_rotation_matrix(v, 1)
[ 1/5*sqrt(5)  2/5*sqrt(5)            0]
[-2/5*sqrt(5)  1/5*sqrt(5)            0]
[           0            0            1]
sage: ith_to_zero_rotation_matrix(v, 2)
[             1              0              0]
[             0  2/13*sqrt(13)  3/13*sqrt(13)]
[             0 -3/13*sqrt(13)  2/13*sqrt(13)]

sage: ith_to_zero_rotation_matrix(v, 0) * v
(0, 2, sqrt(10))
sage: ith_to_zero_rotation_matrix(v, 1) * v
(sqrt(5), 0, 3)
sage: ith_to_zero_rotation_matrix(v, 2) * v
(1, sqrt(13), 0)

Other ring:

sage: ith_to_zero_rotation_matrix(v, 2, ring=RR)
[  1.00000000000000  0.000000000000000  0.000000000000000]
[ 0.000000000000000  0.554700196225229  0.832050294337844]
[ 0.000000000000000 -0.832050294337844  0.554700196225229]
sage: ith_to_zero_rotation_matrix(v, 2, ring=RDF)
[                1.0                 0.0                 0.0]
[                0.0  0.5547001962252291  0.8320502943378437]
[                0.0 -0.8320502943378437  0.5547001962252291]

On the symbolic ring:

sage: x,y,z = var('x,y,z')
sage: v = vector((x,y,z))
sage: ith_to_zero_rotation_matrix(v, 2)
[                 1                  0                  0]
[                 0  y/sqrt(y^2 + z^2)  z/sqrt(y^2 + z^2)]
[                 0 -z/sqrt(y^2 + z^2)  y/sqrt(y^2 + z^2)]
sage: ith_to_zero_rotation_matrix(v, 2) * v
(x, y^2/sqrt(y^2 + z^2) + z^2/sqrt(y^2 + z^2), 0)

AUTHORS:

Sébastien Labbé (April 2010)

sage.matrix.special.jordan_block(eigenvalue, size, sparse=False)#

This function is available as jordan_block(…) and matrix.jordan_block(…).

Return the Jordan block for the given eigenvalue with given size.

INPUT:

eigenvalue – eigenvalue for the diagonal entries of the block
size – size of the square matrix
sparse – (default: False) - if True, return a sparse matrix

EXAMPLES:

sage: jordan_block(5, 3)
[5 1 0]
[0 5 1]
[0 0 5]

sage.matrix.special.lehmer(ring, n=0)#

This function is available as lehmer(…) and matrix.lehmer(…).

Return the \(n \times n\) Lehmer matrix.

The default ring is the rationals.

Element \((i, j)\) in the Lehmer matrix is \(min(i, j)/max(i, j)\).

See Wikipedia article Lehmer_matrix.

EXAMPLES:

sage: matrix.lehmer(3)
[  1 1/2 1/3]
[1/2   1 2/3]
[1/3 2/3   1]

sage.matrix.special.matrix_method(func=None, name=None)#

Allows a function to be tab-completed on the global matrix constructor object.

INPUT:

*function – a single argument. The function that is being decorated.
**kwds – a single optional keyword argument name=<string>. The name of the corresponding method in the global matrix constructor object. If not given, it is derived from the function name.

EXAMPLES:

sage: from sage.matrix.constructor import matrix_method
sage: def foo_matrix(n): return matrix.diagonal(range(n))
sage: matrix_method(foo_matrix)
<function foo_matrix at ...>
sage: matrix.foo(5)
[0 0 0 0 0]
[0 1 0 0 0]
[0 0 2 0 0]
[0 0 0 3 0]
[0 0 0 0 4]
sage: matrix_method(foo_matrix, name='bar')
<function foo_matrix at ...>
sage: matrix.bar(3)
[0 0 0]
[0 1 0]
[0 0 2]

sage.matrix.special.ones_matrix(ring, nrows=None, ncols=None, sparse=False)#

This function is available as ones_matrix(…) and matrix.ones(…).

Return a matrix with all entries equal to 1.

CALL FORMATS:

In each case, the optional keyword sparse can be used.

ones_matrix(ring, nrows, ncols)

ones_matrix(ring, nrows)

ones_matrix(nrows, ncols)

ones_matrix(nrows)

INPUT:

ring - default: ZZ - base ring for the matrix.
nrows - number of rows in the matrix.
ncols - number of columns in the matrix. If omitted, defaults to the number of rows, producing a square matrix.
sparse - default: False - if True creates a sparse representation.

OUTPUT:

A matrix of size nrows by ncols over the ring with every entry equal to 1. While the result is far from sparse, you may wish to choose a sparse representation when mixing this matrix with other sparse matrices.

EXAMPLES:

A call specifying the ring and the size.

sage: M= ones_matrix(QQ, 2, 5); M
[1 1 1 1 1]
[1 1 1 1 1]
sage: M.parent()
Full MatrixSpace of 2 by 5 dense matrices over Rational Field

Without specifying the number of columns, the result is square.

sage: M = ones_matrix(RR, 2); M
[1.00000000000000 1.00000000000000]
[1.00000000000000 1.00000000000000]
sage: M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Real Field with 53 bits of precision

The ring defaults to the integers if not given.

sage: M = ones_matrix(2, 3); M
[1 1 1]
[1 1 1]
sage: M.parent()
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

A lone integer input produces a square matrix over the integers.

sage: M = ones_matrix(3); M
[1 1 1]
[1 1 1]
[1 1 1]
sage: M.parent()
Full MatrixSpace of 3 by 3 dense matrices over Integer Ring

The result can have a sparse implementation.

sage: M = ones_matrix(3, 1, sparse=True); M
[1]
[1]
[1]
sage: M.parent()
Full MatrixSpace of 3 by 1 sparse matrices over Integer Ring

Giving just a ring will yield an error.

sage: ones_matrix(CC)
Traceback (most recent call last):
...
ValueError: constructing an all ones matrix requires at least one dimension

sage.matrix.special.random_diagonalizable_matrix(parent, eigenvalues=None, dimensions=None)#

This function is available as random_diagonalizable_matrix(…) and matrix.random_diagonalizable(…).

Create a random matrix that diagonalizes nicely.

To be used as a teaching tool. Return matrices have only real eigenvalues.

INPUT:

If eigenvalues and dimensions are not specified in a list, they will be assigned randomly.

parent - the desired size of the square matrix.
eigenvalues - the list of desired eigenvalues (default=None).
dimensions - the list of dimensions corresponding to each eigenspace (default=None).

OUTPUT:

A square, diagonalizable, matrix with only integer entries. The eigenspaces of this matrix, if computed by hand, give basis vectors with only integer entries.

Note

It is easiest to use this function via a call to the random_matrix() function with the algorithm='diagonalizable' keyword. We provide one example accessing this function directly, while the remainder will use this more general function.

EXAMPLES:

A diagonalizable matrix, size 5.

sage: from sage.matrix.constructor import random_diagonalizable_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5)
sage: A = random_diagonalizable_matrix(matrix_space)
sage: eigenvalues = A.eigenvalues()
sage: S = A.right_eigenmatrix()[1]
sage: eigenvalues2 = (S.inverse()*A*S).diagonal()
sage: sorted(eigenvalues) == sorted(eigenvalues2)
True

A diagonalizable matrix with eigenvalues and dimensions designated, with a check that if eigenvectors were calculated by hand entries would all be integers.

sage: eigenvalues = [ZZ.random_element() for _ in range(3)]
sage: B = random_matrix(QQ, 6, algorithm='diagonalizable', eigenvalues=eigenvalues, dimensions=[2,3,1])
sage: all(x in ZZ for x in (B-(-12*identity_matrix(6))).rref().list())
True
sage: all(x in ZZ for x in (B-(4*identity_matrix(6))).rref().list())
True
sage: all(x in ZZ for x in (B-(6*identity_matrix(6))).rref().list())
True
sage: S = B.right_eigenmatrix()[1]
sage: eigenvalues2 = (S.inverse()*B*S).diagonal()
sage: all(e in eigenvalues for e in eigenvalues2)
True

Todo

Modify the routine to allow for complex eigenvalues.

AUTHOR:

Billy Wonderly (2010-07)

sage.matrix.special.random_echelonizable_matrix(parent, rank, upper_bound=None, max_tries=100)#

This function is available as random_echelonizable_matrix(…) and matrix.random_echelonizable(…).

Generate a matrix of a desired size and rank, over a desired ring, whose reduced row-echelon form has only integral values.

INPUT:

parent – A matrix space specifying the base ring, dimensions and representation (dense/sparse) for the result. The base ring must be exact.
rank – Rank of result, i.e the number of non-zero rows in the reduced row echelon form.
upper_bound – If designated, size control of the matrix entries is desired. Set upper_bound to 1 more than the maximum value entries can achieve. If None, no size control occurs. But see the warning below. (default: None)
max_tries - If designated, number of tries used to generate each new random row; only matters when upper_bound!=None. Used to prevent endless looping. (default: 100)

OUTPUT:

A matrix not in reduced row-echelon form with the desired dimensions and properties.

Warning

When upper_bound is set, it is possible for this constructor to fail with a ValueError. This may happen when the upper_bound, rank and/or matrix dimensions are all so small that it becomes infeasible or unlikely to create the requested matrix. If you must have this routine return successfully, do not set upper_bound.

Note

It is easiest to use this function via a call to the random_matrix() function with the algorithm='echelonizable' keyword. We provide one example accessing this function directly, while the remainder will use this more general function.

EXAMPLES:

Generated matrices have the desired dimensions, rank and entry size. The matrix in reduced row-echelon form has only integer entries.

sage: from sage.matrix.constructor import random_echelonizable_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5, 6)
sage: A = random_echelonizable_matrix(matrix_space, rank=4, upper_bound=40)
sage: A.rank()
4
sage: max(map(abs,A.list()))<40
True
sage: A.rref() == A.rref().change_ring(ZZ)
True

An example with default settings (i.e. no entry size control).

sage: C=random_matrix(QQ, 6, 7, algorithm='echelonizable', rank=5)
sage: C.rank()
5
sage: C.rref() == C.rref().change_ring(ZZ)
True

A matrix without size control may have very large entry sizes.

sage: D=random_matrix(ZZ, 7, 8, algorithm='echelonizable', rank=6); D  # random
[    1     2     8   -35  -178  -239  -284   778]
[    4     9    37  -163  -827 -1111 -1324  3624]
[    5     6    21   -88  -454  -607  -708  1951]
[   -4    -5   -22    97   491   656   779 -2140]
[    4     4    13   -55  -283  -377  -436  1206]
[    4    11    43  -194  -982 -1319 -1576  4310]
[   -1    -2   -13    59   294   394   481 -1312]

Matrices can be generated over any exact ring.

sage: F.<a>=GF(2^3)
sage: B = random_matrix(F, 4, 5, algorithm='echelonizable', rank=4, upper_bound=None)
sage: B.rank()
4
sage: B.base_ring() is F
True

Square matrices over ZZ or QQ with full rank are always unimodular.

sage: E = random_matrix(QQ, 7, 7, algorithm='echelonizable', rank=7)
sage: det(E)
1
sage: E = random_matrix(ZZ, 7, 7, algorithm='echelonizable', rank=7)
sage: det(E)
1

AUTHOR:

Billy Wonderly (2010-07)

sage.matrix.special.random_matrix(ring, nrows, ncols=None, algorithm='randomize', implementation=None, *args, **kwds)#

This function is available as random_matrix(…) and matrix.random(…).

Return a random matrix with entries in a specified ring, and possibly with additional properties.

INPUT:

ring – base ring for entries of the matrix
nrows – Integer; number of rows
ncols – (default: None); number of columns; if None defaults to nrows
algorithm – (default: randomize); determines what properties the matrix will have. See examples below for possible additional arguments.
- randomize – create a matrix of random elements from the base ring, possibly controlling the density of non-zero entries.
- echelon_form – creates a matrix in echelon form
- echelonizable – creates a matrix that has a predictable echelon form
- subspaces – creates a matrix whose four subspaces, when explored, have reasonably sized, integral valued, entries.
- unimodular – creates a matrix of determinant 1.
- diagonalizable – creates a diagonalizable matrix whose eigenvectors, if computed by hand, will have only integer entries.
implementation – (None or string or a matrix class) a possible implementation. See the documentation of the constructor of MatrixSpace.
*args, **kwds – arguments and keywords to describe additional properties. See more detailed documentation below.

Warning

Matrices generated are not uniformly distributed. For unimodular matrices over finite field this function does not even generate all of them: for example Matrix.random(GF(3), 2, algorithm='unimodular') never generates [[2,0],[0,2]]. This function is made for teaching purposes.

Warning

An upper bound on the absolute value of the entries may be set when the algorithm is echelonizable or unimodular. In these cases it is possible for this constructor to fail with a ValueError. If you must have this routine return successfully, do not set upper_bound. This behavior can be partially controlled by a max_tries keyword.

Note

When constructing matrices with random entries and no additional properties (i.e. when algorithm='randomize'), most of the randomness is controlled by the random_element method for elements of the base ring of the matrix, so the documentation of that method may be relevant or useful.

EXAMPLES:

Random integer matrices. With no arguments, the majority of the entries are zero, -1, and 1, and rarely “large.”

sage: from collections import defaultdict
sage: total_count = 0
sage: dic = defaultdict(Integer)
sage: def add_samples(*args, **kwds):
....:     global dic, total_count
....:     for _ in range(100):
....:         A = random_matrix(*args, **kwds)
....:         for a in A.list():
....:             dic[a] += 1
....:             total_count += 1.0

sage: expected = lambda n : 2 / (5*abs(n)*(abs(n) + 1)) if n != 0 else 1/5
sage: expected(0)
1/5
sage: expected(0) == expected(1) == expected(-1)
True
sage: expected(100)
1/25250
sage: add_samples(ZZ, 5, 5)
sage: while not all(abs(dic[a]/total_count - expected(a)) < 0.001 for a in dic):
....:     add_samples(ZZ, 5, 5)

The distribution keyword set to uniform will limit values between -2 and 2.

sage: expected = lambda n : 1/5 if n in range(-2, 3) else 0
sage: total_count = 0
sage: dic = defaultdict(Integer)
sage: add_samples(ZZ, 5, 5, distribution='uniform')
sage: while not all(abs(dic[a]/total_count - expected(a)) < 0.001 for a in dic):
....:     add_samples(ZZ, 5, 5, distribution='uniform')

The x and y keywords can be used to distribute entries uniformly. When both are used x is the minimum and y is one greater than the maximum.

sage: expected = lambda n : 1/30 if n in range(70, 100) else 0
sage: total_count = 0
sage: dic = defaultdict(Integer)
sage: add_samples(ZZ, 4, 8, x=70, y=100)
sage: while not all(abs(dic[a]/total_count - expected(a)) < 0.001 for a in dic):
....:     add_samples(ZZ, 4, 8, x=70, y=100)

sage: expected = lambda n : 1/10 if n in range(-5, 5) else 0
sage: total_count = 0
sage: dic = defaultdict(Integer)
sage: add_samples(ZZ, 3, 7, x=-5, y=5)
sage: while not all(abs(dic[a]/total_count - expected(a)) < 0.001 for a in dic):
....:     add_samples(ZZ, 3, 7, x=-5, y=5)

If only x is given, then it is used as the upper bound of a range starting at 0.

sage: expected = lambda n : 1/25 if n in range(25) else 0
sage: total_count = 0
sage: dic = defaultdict(Integer)
sage: add_samples(ZZ, 5, 5, x=25)
sage: while not all(abs(dic[a]/total_count - expected(a)) < 0.001 for a in dic):
....:     add_samples(ZZ, 5, 5, x=25)

To control the number of nonzero entries, use the density keyword at a value strictly below the default of 1.0. The density keyword is used to compute the number of entries per row that will be nonzero, but the same entry may be selected more than once. So the value provided will be an upper bound for the density of the created matrix. Note that for a square matrix it is only necessary to set a single dimension.

sage: def add_sample(*args, **kwds):
....:     global density_sum, total_count
....:     total_count += 1.0
....:     A = random_matrix(*args, **kwds)
....:     density_sum += float(A.density())

sage: density_sum = 0.0
sage: total_count = 0.0
sage: add_sample(ZZ, 5, x=-10, y=10, density=0.75)
sage: expected_density = (1 - (4/5)^3)
sage: float(expected_density)
0.488
sage: while abs(density_sum/total_count - expected_density) > 0.001:
....:     add_sample(ZZ, 5, x=-10, y=10, density=0.75)

sage: density_sum = 0.0
sage: total_count = 0.0
sage: add_sample(ZZ, 5, x=20, y=30, density=0.75)
sage: while abs(density_sum/total_count - expected_density) > 0.001:
....:     add_sample(ZZ, 5, x=20, y=30, density=0.75)

sage: density_sum = 0.0
sage: total_count = 0.0
sage: add_sample(ZZ, 100, x=20, y=30, density=0.75)
sage: expected_density = (1 - (99/100)^75)
sage: float(expected_density)
0.529...
sage: while abs(density_sum/total_count - expected_density) > 0.001:
....:     add_sample(ZZ, 100, x=20, y=30, density=0.75)

For a matrix with low density it may be advisable to insist on a sparse representation, as this representation is not selected automatically.

sage: A=random_matrix(ZZ, 5, 5)
sage: A.is_sparse()
False
sage: A = random_matrix(ZZ, 5, 5, sparse=True)
sage: A.is_sparse()
True

For algorithm testing you might want to control the number of bits, say 10,000 entries, each limited to 16 bits.

sage: A = random_matrix(ZZ, 100, 100, x=2^16); A
100 x 100 dense matrix over Integer Ring (use the '.str()' method to see the entries)

One can prescribe a specific matrix implementation:

sage: K.<a> = FiniteField(2^8)
sage: type(random_matrix(K, 2, 5))
<class 'sage.matrix.matrix_gf2e_dense.Matrix_gf2e_dense'>
sage: type(random_matrix(K, 2, 5, implementation="generic"))
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>

Random rational matrices. Now num_bound and den_bound control the generation of random elements, by specifying limits on the absolute value of numerators and denominators (respectively). Entries will be positive and negative (map the absolute value function through the entries to get all positive values). If either the numerator or denominator bound (or both) is not used, then the values default to 2:

sage: A = random_matrix(QQ, 2, 8, num_bound=20, den_bound=4)
sage: A.dimensions()
(2, 8)
sage: type(A)
<class 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'>
sage: all(a.numerator() in range(-20, 21) and
....:     a.denominator() in range(1, 5)
....:     for a in A.list())
True

sage: A = random_matrix(QQ, 4, density = 0.5, sparse=True)
sage: type(A)
<class 'sage.matrix.matrix_rational_sparse.Matrix_rational_sparse'>
sage: A.density() <= 0.5
True

sage: A = random_matrix(QQ, 3, 10, num_bound = 99, den_bound = 99)
sage: positives = list(map(abs, A.list()))
sage: A1 = matrix(QQ, 3, 10, positives)
sage: all(abs(A.list()[i]) == A1.list()[i] for i in range(30))
True
sage: all(a.numerator() in range(100) and
....:     a.denominator() in range(1, 100)
....:     for a in A1.list())
True

sage: A = random_matrix(QQ, 4, 10, den_bound = 10)
sage: all(a.numerator() in range(-2, 3) and
....:     a.denominator() in range(1, 11)
....:     for a in A.list())
True

sage: A = random_matrix(QQ, 4, 10)
sage: all(a.numerator() in range(-2, 3) and
....:     a.denominator() in range(1, 3)
....:     for a in A.list())
True

Random matrices over other rings. Several classes of matrices have specialized randomize() methods. You can locate these with the Sage command:

search_def('randomize')

The default implementation of randomize() relies on the random_element() method for the base ring. The density and sparse keywords behave as described above. Since we have a different randomisation when using the optional meataxe package, we have to make sure that we use the default implementation in this test:

sage: K.<a>=FiniteField(3^2)
sage: A = random_matrix(K, 2, 5, implementation='generic')
sage: type(A)
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>
sage: A.base_ring() is K
True
sage: TestSuite(A).run()

sage: A = random_matrix(RR, 3, 4, density=0.66)
sage: type(A)
<class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>
sage: A.base_ring() is RR
True
sage: TestSuite(A).run()

sage: A = random_matrix(ComplexField(32), 3, density=0.8, sparse=True)
sage: A.is_sparse()
True
sage: type(A)
<class 'sage.matrix.matrix_generic_sparse.Matrix_generic_sparse'>
sage: A.base_ring() is ComplexField(32)
True
sage: TestSuite(A).run()

Random matrices in echelon form. The algorithm='echelon_form' keyword, along with a requested number of non-zero rows (num_pivots) will return a random matrix in echelon form. When the base ring is QQ the result has integer entries. Other exact rings may be also specified.

sage: A = random_matrix(QQ, 4, 8, algorithm='echelon_form', num_pivots=3)
sage: A.base_ring()
Rational Field
sage: (A.nrows(), A.ncols())
(4, 8)
sage: A in sage.matrix.matrix_space.MatrixSpace(ZZ, 4, 8)
True
sage: A.rank()
3
sage: A == A.rref()
True

For more, see the documentation of the random_rref_matrix() function. In the notebook or at the Sage command-line, first execute the following to make this further documentation available:

from sage.matrix.constructor import random_rref_matrix

Random matrices with predictable echelon forms. The algorithm='echelonizable' keyword, along with a requested rank (rank) and optional size control (upper_bound) will return a random matrix in echelon form. When the base ring is ZZ or QQ the result has integer entries, whose magnitudes can be limited by the value of upper_bound, and the echelon form of the matrix also has integer entries. Other exact rings may be also specified, but there is no notion of controlling the size. Square matrices of full rank generated by this function always have determinant one, and can be constructed with the unimodular keyword.

sage: A = random_matrix(QQ, 4, 8, algorithm='echelonizable', rank=3, upper_bound=60)
sage: A.base_ring()
Rational Field
sage: (A.nrows(), A.ncols())
(4, 8)
sage: A in sage.matrix.matrix_space.MatrixSpace(ZZ, 4, 8)
True
sage: A.rank()
3
sage: all(abs(x)<60 for x in A.list())
True
sage: A.rref() in sage.matrix.matrix_space.MatrixSpace(ZZ, 4, 8)
True

For more, see the documentation of the random_echelonizable_matrix() function. In the notebook or at the Sage command-line, first execute the following to make this further documentation available:

from sage.matrix.constructor import random_echelonizable_matrix

Random diagonalizable matrices. The algorithm='diagonalizable' keyword, along with a requested matrix size (size) and optional lists of eigenvalues (eigenvalues) and the corresponding eigenspace dimensions (dimensions) will return a random diagonalizable matrix. When the eigenvalues and dimensions are not specified the result will have randomly generated values for both that fit with the designated size.

sage: A = random_matrix(QQ, 5, algorithm='diagonalizable', eigenvalues=[2,3,-1], dimensions=[1,2,2]); A # random
sage: all(x in ZZ for x in (A-(2*identity_matrix(5))).rref().list())
True
sage: all(x in ZZ for x in (A-(3*identity_matrix(5))).rref().list())
True
sage: all(x in ZZ for x in (A-(-1*identity_matrix(5))).rref().list())
True
sage: A.jordan_form()
[ 2| 0| 0| 0| 0]
[--+--+--+--+--]
[ 0| 3| 0| 0| 0]
[--+--+--+--+--]
[ 0| 0| 3| 0| 0]
[--+--+--+--+--]
[ 0| 0| 0|-1| 0]
[--+--+--+--+--]
[ 0| 0| 0| 0|-1]

For more, see the documentation of the random_diagonalizable_matrix() function. In the notebook or at the Sage command-line, first execute the following to make this further documentation available:

from sage.matrix.constructor import random_diagonalizable_matrix

Random matrices with predictable subspaces. The algorithm='subspaces' keyword, along with an optional rank (rank) will return a matrix whose natural basis vectors for its four fundamental subspaces, if computed as described in the documentation of the random_subspaces_matrix() contain only integer entries. If rank, is not set, the rank of the matrix will be generated randomly.

sage: B = random_matrix(QQ, 5, 6, algorithm='subspaces', rank=3); B #random
sage: B_expanded=B.augment(identity_matrix(5)).rref()
sage: (B.nrows(), B.ncols())
(5, 6)
sage: all(x in ZZ for x in B_expanded.list())
True
sage: C=B_expanded.submatrix(0,0,B.nrows()-B.nullity(),B.ncols())
sage: L=B_expanded.submatrix(B.nrows()-B.nullity(),B.ncols())
sage: B.right_kernel() == C.right_kernel()
True
sage: B.row_space() == C.row_space()
True
sage: B.column_space() == L.right_kernel()
True
sage: B.left_kernel() == L.row_space()
True

For more, see the documentation of the random_subspaces_matrix() function. In the notebook or at the Sage command-line, first execute the following to make this further documentation available:

from sage.matrix.constructor import random_subspaces_matrix

Random unimodular matrices. The algorithm='unimodular' keyword, along with an optional entry size control (upper_bound) will return a matrix of determinant 1. When the base ring is ZZ or QQ the result has integer entries, whose magnitudes can be limited by the value of upper_bound.

sage: C=random_matrix(QQ, 5, algorithm='unimodular', upper_bound=70); C # random
sage: det(C)
1
sage: C.base_ring()
Rational Field
sage: (C.nrows(), C.ncols())
(5, 5)
sage: all(abs(x)<70 for x in C.list())
True

For more, see the documentation of the random_unimodular_matrix() function. In the notebook or at the Sage command-line, first execute the following to make this further documentation available:

from sage.matrix.constructor import random_unimodular_matrix

AUTHOR:

William Stein (2007-02-06)
Rob Beezer (2010-08-25) Documentation, code to allow additional types of output

sage.matrix.special.random_rref_matrix(parent, num_pivots)#

This function is available as random_rref_matrix(…) and matrix.random_rref(…).

Generate a matrix in reduced row-echelon form with a specified number of non-zero rows.

INPUT:

parent – A matrix space specifying the base ring, dimensions and representation (dense/sparse) for the result. The base ring must be exact.
num_pivots – The number of non-zero rows in the result, i.e. the rank.

OUTPUT:

A matrix in reduced row echelon form with num_pivots non-zero rows. If the base ring is \(ZZ\) or \(QQ\) then the entries are all integers.

Note

It is easiest to use this function via a call to the random_matrix() function with the algorithm='echelon_form' keyword. We provide one example accessing this function directly, while the remainder will use this more general function.

EXAMPLES:

Matrices generated are in reduced row-echelon form with specified rank. If the base ring is \(QQ\) the result has only integer entries.

sage: from sage.matrix.constructor import random_rref_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5, 6)
sage: A = random_rref_matrix(matrix_space, num_pivots=4); A # random
[ 1  0  0 -6  0 -3]
[ 0  1  0  2  0  3]
[ 0  0  1 -4  0 -2]
[ 0  0  0  0  1  3]
[ 0  0  0  0  0  0]
sage: A.base_ring()
Rational Field
sage: (A.nrows(), A.ncols())
(5, 6)
sage: A in sage.matrix.matrix_space.MatrixSpace(ZZ, 5, 6)
True
sage: A.rank()
4
sage: A == A.rref()
True

Matrices can be generated over other exact rings.

sage: B = random_matrix(FiniteField(7), 4, 4, algorithm='echelon_form', num_pivots=3); B # random
[1 0 0 0]
[0 1 0 6]
[0 0 1 4]
[0 0 0 0]
sage: B.rank() == 3
True
sage: B.base_ring()
Finite Field of size 7
sage: B == B.rref()
True

AUTHOR:

Billy Wonderly (2010-07)

sage.matrix.special.random_subspaces_matrix(parent, rank=None)#

This function is available as random_subspaces_matrix(…) and matrix.random_subspaces(…).

Create a matrix of the designated size and rank whose right and left null spaces, column space, and row space have desirable properties that simplify the subspaces.

INPUT:

parent - A matrix space specifying the base ring, dimensions, and representation (dense/sparse) for the result. The base ring must be exact.
rank - The desired rank of the return matrix (default: None).

OUTPUT:

A matrix whose natural basis vectors for its four subspaces, when computed, have reasonably sized, integral valued, entries.

Note

It is easiest to use this function via a call to the random_matrix() function with the algorithm='subspaces' keyword. We provide one example accessing this function directly, while the remainder will use this more general function.

EXAMPLES:

A 6x8 matrix with designated rank of 3. The four subspaces are determined using one simple routine in which we augment the original matrix with the equal row dimension identity matrix. The resulting matrix is then put in reduced row-echelon form and the subspaces can then be determined by analyzing subdivisions of this matrix. See the four subspaces routine in [Bee] for more.

sage: from sage.matrix.constructor import random_subspaces_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 6, 8)
sage: B = random_subspaces_matrix(matrix_space, rank=3)
sage: B.rank()
3
sage: B.nullity()
3
sage: (B.nrows(), B.ncols())
(6, 8)
sage: all(x in ZZ for x in B.list())
True
sage: B_expanded = B.augment(identity_matrix(6)).rref()
sage: all(x in ZZ for x in B_expanded.list())
True

Check that we fixed trac ticket #10543 (echelon forms should be immutable):

sage: B_expanded.is_immutable()
True

We want to modify B_expanded, so replace it with a copy:

sage: B_expanded = copy(B_expanded)
sage: B_expanded.subdivide(B.nrows()-B.nullity(), B.ncols())
sage: C = B_expanded.subdivision(0, 0)
sage: L = B_expanded.subdivision(1, 1)
sage: B.right_kernel() == C.right_kernel()
True
sage: B.row_space() == C.row_space()
True
sage: B.column_space() == L.right_kernel()
True
sage: B.left_kernel() == L.row_space()
True

A matrix to show that the null space of the L matrix is the column space of the starting matrix.

sage: A = random_matrix(QQ, 5, 7, algorithm='subspaces', rank=None)
sage: (A.nrows(), A.ncols())
(5, 7)
sage: all(x in ZZ for x in A.list())
True
sage: A_expanded=A.augment(identity_matrix(5)).rref()
sage: all(x in ZZ for x in A_expanded.list())
True
sage: C = A_expanded.submatrix(0,0,A.nrows()-A.nullity(), A.ncols())
sage: L = A_expanded.submatrix(A.nrows()-A.nullity(), A.ncols())
sage: A.right_kernel() == C.right_kernel()
True
sage: A.row_space() == C.row_space()
True
sage: A.column_space() == L.right_kernel()
True
sage: A.left_kernel() == L.row_space()
True

AUTHOR:

Billy Wonderly (2010-07)

sage.matrix.special.random_unimodular_matrix(parent, upper_bound=None, max_tries=100)#

This function is available as random_unimodular_matrix(…) and matrix.random_unimodular(…).

Generate a random unimodular (determinant 1) matrix of a desired size over a desired ring.

INPUT:

parent - A matrix space specifying the base ring, dimensions and representation (dense/sparse) for the result. The base ring must be exact.
upper_bound - For large matrices over QQ or ZZ, upper_bound is the largest value matrix entries can achieve. But see the warning below.
max_tries - If designated, number of tries used to generate each new random row; only matters when upper_bound!=None. Used to prevent endless looping. (default: 100)

A matrix not in reduced row-echelon form with the desired dimensions and properties.

OUTPUT:

An invertible matrix with the desired properties and determinant 1.

Warning

Note

It is easiest to use this function via a call to the random_matrix() function with the algorithm='unimodular' keyword. We provide one example accessing this function directly, while the remainder will use this more general function.

EXAMPLES:

A matrix size 5 over QQ.

sage: from sage.matrix.constructor import random_unimodular_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5)
sage: A = random_unimodular_matrix(matrix_space)
sage: det(A)
1

A matrix size 6 with entries no larger than 50.

sage: B = random_matrix(ZZ, 7, algorithm='unimodular', upper_bound=50)
sage: det(B)
1
sage: all(abs(b) < 50 for b in B.list())
True

A matrix over the number Field in \(y\) with defining polynomial \(y^2-2y-2\).

sage: y = var('y')
sage: K = NumberField(y^2-2*y-2, 'y')
sage: C = random_matrix(K, 3, algorithm='unimodular')
sage: det(C)
1
sage: C.base_ring() is K
True

AUTHOR:

Billy Wonderly (2010-07)

sage.matrix.special.toeplitz(c, r, ring=None)#

This function is available as toeplitz(…) and matrix.toeplitz(…).

Return a Toeplitz matrix of given first column and first row.

In a Toeplitz matrix, each descending diagonal from left to right is constant, such that:

\[T_{i,j} = T_{i+1, j+1}.\]

For more information see the Wikipedia article Toeplitz_matrix.

INPUT:

c – vector, first column of the Toeplitz matrix
r – vector, first row of the Toeplitz matrix, counting from the second column
ring – base ring (optional, default: None) of the resulting matrix

EXAMPLES:

A rectangular Toeplitz matrix:

sage: matrix.toeplitz([1..4], [5..6])
[1 5 6]
[2 1 5]
[3 2 1]
[4 3 2]

The following \(N\times N\) Toeplitz matrix arises in the discretization of boundary value problems:

sage: N = 4
sage: matrix.toeplitz([-2, 1] + [0]*(N-2), [1] + [0]*(N-2))
[-2  1  0  0]
[ 1 -2  1  0]
[ 0  1 -2  1]
[ 0  0  1 -2]

sage.matrix.special.vandermonde(v, ring=None)#

This function is available as vandermonde(…) and matrix.vandermonde(…).

Return a Vandermonde matrix of the given vector.

The \(n\) dimensional Vandermonde matrix is a square matrix with columns being the powers of a given vector \(v\),

\[V_{ij} = v_i^{j-1},\qquad i, j = 1,\ldots, n.\]

For more information see the Wikipedia article Vandermonde_matrix.

INPUT:

v – vector, the second column of the Vandermonde matrix
ring – base ring (optional, default: None) of the resulting matrix

EXAMPLES:

A Vandermonde matrix of order three over the symbolic ring:

sage: matrix.vandermonde(SR.var(['x0', 'x1', 'x2']))
[   1   x0 x0^2]
[   1   x1 x1^2]
[   1   x2 x2^2]

sage.matrix.special.vector_on_axis_rotation_matrix(v, i, ring=None)#

This function is available as vector_on_axis_rotation_matrix(…) and matrix.vector_on_axis_rotation(…).

Return a rotation matrix \(M\) such that \(det(M)=1\) sending the vector \(v\) on the i-th axis so that all other coordinates of \(Mv\) are zero.

Note

Such a matrix is not uniquely determined. This function returns one such matrix.

INPUT:

v – vector
i – integer
ring – ring (optional, default: None) of the resulting matrix

OUTPUT:

A matrix

EXAMPLES:

sage: from sage.matrix.constructor import vector_on_axis_rotation_matrix
sage: v = vector((1,2,3))
sage: vector_on_axis_rotation_matrix(v, 2) * v
(0, 0, sqrt(14))
sage: vector_on_axis_rotation_matrix(v, 1) * v
(0, sqrt(14), 0)
sage: vector_on_axis_rotation_matrix(v, 0) * v
(sqrt(14), 0, 0)

sage: x,y = var('x,y')
sage: v = vector((x,y))
sage: vector_on_axis_rotation_matrix(v, 1)
[ y/sqrt(x^2 + y^2) -x/sqrt(x^2 + y^2)]
[ x/sqrt(x^2 + y^2)  y/sqrt(x^2 + y^2)]
sage: vector_on_axis_rotation_matrix(v, 0)
[ x/sqrt(x^2 + y^2)  y/sqrt(x^2 + y^2)]
[-y/sqrt(x^2 + y^2)  x/sqrt(x^2 + y^2)]
sage: vector_on_axis_rotation_matrix(v, 0) * v
(x^2/sqrt(x^2 + y^2) + y^2/sqrt(x^2 + y^2), 0)
sage: vector_on_axis_rotation_matrix(v, 1) * v
(0, x^2/sqrt(x^2 + y^2) + y^2/sqrt(x^2 + y^2))

sage: v = vector((1,2,3,4))
sage: vector_on_axis_rotation_matrix(v, 0) * v
(sqrt(30), 0, 0, 0)
sage: vector_on_axis_rotation_matrix(v, 0, ring=RealField(10))
[ 0.18  0.37  0.55  0.73]
[-0.98 0.068  0.10  0.14]
[ 0.00 -0.93  0.22  0.30]
[ 0.00  0.00 -0.80  0.60]
sage: vector_on_axis_rotation_matrix(v, 0, ring=RealField(10)) * v
(5.5, 0.00..., 0.00..., 0.00...)

AUTHORS:

Sébastien Labbé (April 2010)

sage.matrix.special.zero_matrix(ring, nrows=None, ncols=None, sparse=False)#

This function is available as zero_matrix(…) and matrix.zero(…).

Return the \(nrows \times ncols\) zero matrix over the given ring.

The default ring is the integers.

EXAMPLES:

sage: M = zero_matrix(QQ, 2); M
[0 0]
[0 0]
sage: M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: M = zero_matrix(2, 3); M
[0 0 0]
[0 0 0]
sage: M.parent()
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: M.is_mutable()
True
sage: M = zero_matrix(3, 1, sparse=True); M
[0]
[0]
[0]
sage: M.parent()
Full MatrixSpace of 3 by 1 sparse matrices over Integer Ring
sage: M.is_mutable()
True
sage: matrix.zero(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]