General matrix Constructor and display options

sage.matrix.constructor.Matrix(*args, **kwds)[source]

Create a matrix.

This implements the matrix constructor:

sage: matrix([[1,2],[3,4]])
[1 2]
[3 4]

>>> from sage.all import *
>>> matrix([[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
[1 2]
[3 4]

matrix([[1,2],[3,4]])

It also contains methods to create special types of matrices, see matrix.[tab] for more options. For example:

sage: matrix.identity(2)
[1 0]
[0 1]

>>> from sage.all import *
>>> matrix.identity(Integer(2))
[1 0]
[0 1]

matrix.identity(2)

INPUT:

The matrix() command takes the entries of a matrix, optionally preceded by a ring and the dimensions of the matrix, and returns a matrix.

The entries of a matrix can be specified as a flat list of elements, a list of lists (i.e., a list of rows), a list of Sage vectors, a callable object, or a dictionary having positions as keys and matrix entries as values (see the examples). If you pass in a callable object, then you must specify the number of rows and columns. You can create a matrix of zeros by passing an empty list or the integer zero for the entries. To construct a multiple of the identity (\(cI\)), you can specify square dimensions and pass in \(c\). Calling matrix() with a Sage object may return something that makes sense. Calling matrix() with a NumPy array will convert the array to a matrix.

All arguments (even the positional ones) are optional.

Positional and keyword arguments:

  • base_ring – parent of the entries of the matrix (despite the name, this is not a priori required to be a ring). By default, determine this from the given entries, falling back to ZZ if no entries are given.

  • nrows – the number of rows in the matrix, or a finite or enumerated family of arbitrary objects that index the rows of the matrix

  • ncols – the number of columns in the matrix, or a finite or enumerated family of arbitrary objects that index the columns of the matrix

  • entries – see examples below

If any of nrows, ncols, row_keys, column_keys is given as keyword argument, then none of these may be given as positional arguments.

Keyword-only arguments:

  • sparse – boolean; create a sparse matrix. This defaults to True when the entries are given as a dictionary, otherwise defaults to False.

  • row_keys – a finite or enumerated family of arbitrary objects that index the rows of the matrix

  • column_keys – a finite or enumerated family of arbitrary objects that index the columns of the matrix

  • space – matrix space which will be the parent of the output matrix. This determines base_ring, nrows, row_keys, ncols, column_keys, and sparse.

  • immutable – boolean; make the matrix immutable. By default, the output matrix is mutable.

OUTPUT: a matrix or, more generally, a homomorphism between free modules

EXAMPLES:

sage: m = matrix(2); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

>>> from sage.all import *
>>> m = matrix(Integer(2)); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

m = matrix(2); m; m.parent()

sage: m = matrix(2, 3); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

>>> from sage.all import *
>>> m = matrix(Integer(2), Integer(3)); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

m = matrix(2, 3); m; m.parent()

sage: m = matrix(QQ, [[1,2,3], [4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, [[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

m = matrix(QQ, [[1,2,3], [4,5,6]]); m; m.parent()

sage: m = matrix(QQ, 3, 3, lambda i, j: i + j); m
[0 1 2]
[1 2 3]
[2 3 4]
sage: m = matrix(3, lambda i, j: i - j); m
[ 0 -1 -2]
[ 1  0 -1]
[ 2  1  0]

>>> from sage.all import *
>>> m = matrix(QQ, Integer(3), Integer(3), lambda i, j: i + j); m
[0 1 2]
[1 2 3]
[2 3 4]
>>> m = matrix(Integer(3), lambda i, j: i - j); m
[ 0 -1 -2]
[ 1  0 -1]
[ 2  1  0]

m = matrix(QQ, 3, 3, lambda i, j: i + j); m
m = matrix(3, lambda i, j: i - j); m

sage: matrix(QQ, 2, 3, lambda x, y: x + y)
[0 1 2]
[1 2 3]
sage: matrix(QQ, 5, 5, lambda x, y: (x+1) / (y+1))
[  1 1/2 1/3 1/4 1/5]
[  2   1 2/3 1/2 2/5]
[  3 3/2   1 3/4 3/5]
[  4   2 4/3   1 4/5]
[  5 5/2 5/3 5/4   1]

>>> from sage.all import *
>>> matrix(QQ, Integer(2), Integer(3), lambda x, y: x + y)
[0 1 2]
[1 2 3]
>>> matrix(QQ, Integer(5), Integer(5), lambda x, y: (x+Integer(1)) / (y+Integer(1)))
[  1 1/2 1/3 1/4 1/5]
[  2   1 2/3 1/2 2/5]
[  3 3/2   1 3/4 3/5]
[  4   2 4/3   1 4/5]
[  5 5/2 5/3 5/4   1]

matrix(QQ, 2, 3, lambda x, y: x + y)
matrix(QQ, 5, 5, lambda x, y: (x+1) / (y+1))

sage: v1 = vector((1,2,3))
sage: v2 = vector((4,5,6))
sage: m = matrix([v1,v2]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

>>> from sage.all import *
>>> v1 = vector((Integer(1),Integer(2),Integer(3)))
>>> v2 = vector((Integer(4),Integer(5),Integer(6)))
>>> m = matrix([v1,v2]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

v1 = vector((1,2,3))
v2 = vector((4,5,6))
m = matrix([v1,v2]); m; m.parent()

sage: m = matrix(QQ, 2, [1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, Integer(2), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

m = matrix(QQ, 2, [1,2,3,4,5,6]); m; m.parent()

sage: m = matrix(QQ, 2, 3, [1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, Integer(2), Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

m = matrix(QQ, 2, 3, [1,2,3,4,5,6]); m; m.parent()

sage: m = matrix({(0,1): 2, (1,1): 2/5}); m; m.parent()
[  0   2]
[  0 2/5]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

>>> from sage.all import *
>>> m = matrix({(Integer(0),Integer(1)): Integer(2), (Integer(1),Integer(1)): Integer(2)/Integer(5)}); m; m.parent()
[  0   2]
[  0 2/5]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

m = matrix({(0,1): 2, (1,1): 2/5}); m; m.parent()

sage: m = matrix(QQ, 2, 3, {(1,1): 2}); m; m.parent()
[0 0 0]
[0 2 0]
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, Integer(2), Integer(3), {(Integer(1),Integer(1)): Integer(2)}); m; m.parent()
[0 0 0]
[0 2 0]
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

m = matrix(QQ, 2, 3, {(1,1): 2}); m; m.parent()

sage: import numpy                                                              # needs numpy
sage: n = numpy.array([[1,2], [3,4]], float)                                    # needs numpy
sage: m = matrix(n); m; m.parent()                                              # needs numpy
[1.0 2.0]
[3.0 4.0]
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

>>> from sage.all import *
>>> import numpy                                                              # needs numpy
>>> n = numpy.array([[Integer(1),Integer(2)], [Integer(3),Integer(4)]], float)                                    # needs numpy
>>> m = matrix(n); m; m.parent()                                              # needs numpy
[1.0 2.0]
[3.0 4.0]
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

import numpy                                                              # needs numpy
n = numpy.array([[1,2], [3,4]], float)                                    # needs numpy
m = matrix(n); m; m.parent()                                              # needs numpy

sage: v = vector(ZZ, [1, 10, 100])
sage: m = matrix(v); m; m.parent()
[  1  10 100]
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
sage: m = matrix(GF(7), v); m; m.parent()
[1 3 2]
Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7
sage: m = matrix(GF(7), 3, 1, v); m; m.parent()
[1]
[3]
[2]
Full MatrixSpace of 3 by 1 dense matrices over Finite Field of size 7

>>> from sage.all import *
>>> v = vector(ZZ, [Integer(1), Integer(10), Integer(100)])
>>> m = matrix(v); m; m.parent()
[  1  10 100]
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
>>> m = matrix(GF(Integer(7)), v); m; m.parent()
[1 3 2]
Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7
>>> m = matrix(GF(Integer(7)), Integer(3), Integer(1), v); m; m.parent()
[1]
[3]
[2]
Full MatrixSpace of 3 by 1 dense matrices over Finite Field of size 7

v = vector(ZZ, [1, 10, 100])
m = matrix(v); m; m.parent()
m = matrix(GF(7), v); m; m.parent()
m = matrix(GF(7), 3, 1, v); m; m.parent()

sage: matrix(pari.mathilbert(3))                                                # needs sage.libs.pari
[  1 1/2 1/3]
[1/2 1/3 1/4]
[1/3 1/4 1/5]

>>> from sage.all import *
>>> matrix(pari.mathilbert(Integer(3)))                                                # needs sage.libs.pari
[  1 1/2 1/3]
[1/2 1/3 1/4]
[1/3 1/4 1/5]

matrix(pari.mathilbert(3))                                                # needs sage.libs.pari

sage: g = graphs.PetersenGraph()                                                # needs sage.graphs
sage: m = matrix(g); m; m.parent()                                              # needs sage.graphs
[0 1 0 0 1 1 0 0 0 0]
[1 0 1 0 0 0 1 0 0 0]
[0 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 0 0 0 1 0]
[1 0 0 1 0 0 0 0 0 1]
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 1 0 0 0 1]
[0 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 0]
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring

>>> from sage.all import *
>>> g = graphs.PetersenGraph()                                                # needs sage.graphs
>>> m = matrix(g); m; m.parent()                                              # needs sage.graphs
[0 1 0 0 1 1 0 0 0 0]
[1 0 1 0 0 0 1 0 0 0]
[0 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 0 0 0 1 0]
[1 0 0 1 0 0 0 0 0 1]
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 1 0 0 0 1]
[0 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 0]
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring

g = graphs.PetersenGraph()                                                # needs sage.graphs
m = matrix(g); m; m.parent()                                              # needs sage.graphs

sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent()
Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring

>>> from sage.all import *
>>> matrix(ZZ, Integer(10), Integer(10), range(Integer(100)), sparse=True).parent()
Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring

matrix(ZZ, 10, 10, range(100), sparse=True).parent()

sage: R = PolynomialRing(QQ, 9, 'x')
sage: A = matrix(R, 3, 3, R.gens()); A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
sage: det(A)
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(9), 'x')
>>> A = matrix(R, Integer(3), Integer(3), R.gens()); A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
>>> det(A)
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

R = PolynomialRing(QQ, 9, 'x')
A = matrix(R, 3, 3, R.gens()); A
det(A)

sage: M = Matrix([[1,2,3], [4,5,6], [7,8,9]], immutable=True)
sage: M[0] = [9,9,9]
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).

>>> from sage.all import *
>>> M = Matrix([[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)], [Integer(7),Integer(8),Integer(9)]], immutable=True)
>>> M[Integer(0)] = [Integer(9),Integer(9),Integer(9)]
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).

M = Matrix([[1,2,3], [4,5,6], [7,8,9]], immutable=True)
M[0] = [9,9,9]

Using row_keys and column_keys:

sage: M = matrix([[1,2,3], [4,5,6]],
....:            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
  From: Free module generated by {'a', 'b', 'c'} over Integer Ring
  To:   Free module generated by {'u', 'v'} over Integer Ring
sage: print(M._unicode_art_matrix())
  a b c
u⎛1 2 3⎞
v⎝4 5 6⎠

>>> from sage.all import *
>>> M = matrix([[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)]],
...            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
  From: Free module generated by {'a', 'b', 'c'} over Integer Ring
  To:   Free module generated by {'u', 'v'} over Integer Ring
>>> print(M._unicode_art_matrix())
  a b c
u⎛1 2 3⎞
v⎝4 5 6⎠

M = matrix([[1,2,3], [4,5,6]],
           column_keys=['a','b','c'], row_keys=['u','v']); M
print(M._unicode_art_matrix())

It is allowed to specify dimensions redundantly:

sage: M = matrix(2, 3, [[1,2,3], [4,5,6]],
....:            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
From: Free module generated by {'a', 'b', 'c'} over Integer Ring
To:   Free module generated by {'u', 'v'} over Integer Ring

>>> from sage.all import *
>>> M = matrix(Integer(2), Integer(3), [[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)]],
...            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
From: Free module generated by {'a', 'b', 'c'} over Integer Ring
To:   Free module generated by {'u', 'v'} over Integer Ring

M = matrix(2, 3, [[1,2,3], [4,5,6]],
           column_keys=['a','b','c'], row_keys=['u','v']); M

AUTHORS:

  • William Stein: Initial implementation

  • Jason Grout (2008-03): almost a complete rewrite, with bits and pieces from the original implementation

  • Jeroen Demeyer (2016-02-05): major clean up, see Issue #20015 and Issue #20016

  • Jeroen Demeyer (2018-02-20): completely rewritten using MatrixArgs, see Issue #24742

sage.matrix.constructor.matrix(*args, **kwds)[source]

Create a matrix.

This implements the matrix constructor:

sage: matrix([[1,2],[3,4]])
[1 2]
[3 4]

>>> from sage.all import *
>>> matrix([[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
[1 2]
[3 4]

matrix([[1,2],[3,4]])

It also contains methods to create special types of matrices, see matrix.[tab] for more options. For example:

sage: matrix.identity(2)
[1 0]
[0 1]

>>> from sage.all import *
>>> matrix.identity(Integer(2))
[1 0]
[0 1]

matrix.identity(2)

INPUT:

The matrix() command takes the entries of a matrix, optionally preceded by a ring and the dimensions of the matrix, and returns a matrix.

The entries of a matrix can be specified as a flat list of elements, a list of lists (i.e., a list of rows), a list of Sage vectors, a callable object, or a dictionary having positions as keys and matrix entries as values (see the examples). If you pass in a callable object, then you must specify the number of rows and columns. You can create a matrix of zeros by passing an empty list or the integer zero for the entries. To construct a multiple of the identity (\(cI\)), you can specify square dimensions and pass in \(c\). Calling matrix() with a Sage object may return something that makes sense. Calling matrix() with a NumPy array will convert the array to a matrix.

All arguments (even the positional ones) are optional.

Positional and keyword arguments:

  • base_ring – parent of the entries of the matrix (despite the name, this is not a priori required to be a ring). By default, determine this from the given entries, falling back to ZZ if no entries are given.

  • nrows – the number of rows in the matrix, or a finite or enumerated family of arbitrary objects that index the rows of the matrix

  • ncols – the number of columns in the matrix, or a finite or enumerated family of arbitrary objects that index the columns of the matrix

  • entries – see examples below

If any of nrows, ncols, row_keys, column_keys is given as keyword argument, then none of these may be given as positional arguments.

Keyword-only arguments:

  • sparse – boolean; create a sparse matrix. This defaults to True when the entries are given as a dictionary, otherwise defaults to False.

  • row_keys – a finite or enumerated family of arbitrary objects that index the rows of the matrix

  • column_keys – a finite or enumerated family of arbitrary objects that index the columns of the matrix

  • space – matrix space which will be the parent of the output matrix. This determines base_ring, nrows, row_keys, ncols, column_keys, and sparse.

  • immutable – boolean; make the matrix immutable. By default, the output matrix is mutable.

OUTPUT: a matrix or, more generally, a homomorphism between free modules

EXAMPLES:

sage: m = matrix(2); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

>>> from sage.all import *
>>> m = matrix(Integer(2)); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

m = matrix(2); m; m.parent()

sage: m = matrix(2, 3); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

>>> from sage.all import *
>>> m = matrix(Integer(2), Integer(3)); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

m = matrix(2, 3); m; m.parent()

sage: m = matrix(QQ, [[1,2,3], [4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, [[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

m = matrix(QQ, [[1,2,3], [4,5,6]]); m; m.parent()

sage: m = matrix(QQ, 3, 3, lambda i, j: i + j); m
[0 1 2]
[1 2 3]
[2 3 4]
sage: m = matrix(3, lambda i, j: i - j); m
[ 0 -1 -2]
[ 1  0 -1]
[ 2  1  0]

>>> from sage.all import *
>>> m = matrix(QQ, Integer(3), Integer(3), lambda i, j: i + j); m
[0 1 2]
[1 2 3]
[2 3 4]
>>> m = matrix(Integer(3), lambda i, j: i - j); m
[ 0 -1 -2]
[ 1  0 -1]
[ 2  1  0]

m = matrix(QQ, 3, 3, lambda i, j: i + j); m
m = matrix(3, lambda i, j: i - j); m

sage: matrix(QQ, 2, 3, lambda x, y: x + y)
[0 1 2]
[1 2 3]
sage: matrix(QQ, 5, 5, lambda x, y: (x+1) / (y+1))
[  1 1/2 1/3 1/4 1/5]
[  2   1 2/3 1/2 2/5]
[  3 3/2   1 3/4 3/5]
[  4   2 4/3   1 4/5]
[  5 5/2 5/3 5/4   1]

>>> from sage.all import *
>>> matrix(QQ, Integer(2), Integer(3), lambda x, y: x + y)
[0 1 2]
[1 2 3]
>>> matrix(QQ, Integer(5), Integer(5), lambda x, y: (x+Integer(1)) / (y+Integer(1)))
[  1 1/2 1/3 1/4 1/5]
[  2   1 2/3 1/2 2/5]
[  3 3/2   1 3/4 3/5]
[  4   2 4/3   1 4/5]
[  5 5/2 5/3 5/4   1]

matrix(QQ, 2, 3, lambda x, y: x + y)
matrix(QQ, 5, 5, lambda x, y: (x+1) / (y+1))

sage: v1 = vector((1,2,3))
sage: v2 = vector((4,5,6))
sage: m = matrix([v1,v2]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

>>> from sage.all import *
>>> v1 = vector((Integer(1),Integer(2),Integer(3)))
>>> v2 = vector((Integer(4),Integer(5),Integer(6)))
>>> m = matrix([v1,v2]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

v1 = vector((1,2,3))
v2 = vector((4,5,6))
m = matrix([v1,v2]); m; m.parent()

sage: m = matrix(QQ, 2, [1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, Integer(2), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

m = matrix(QQ, 2, [1,2,3,4,5,6]); m; m.parent()

sage: m = matrix(QQ, 2, 3, [1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, Integer(2), Integer(3), [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

m = matrix(QQ, 2, 3, [1,2,3,4,5,6]); m; m.parent()

sage: m = matrix({(0,1): 2, (1,1): 2/5}); m; m.parent()
[  0   2]
[  0 2/5]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

>>> from sage.all import *
>>> m = matrix({(Integer(0),Integer(1)): Integer(2), (Integer(1),Integer(1)): Integer(2)/Integer(5)}); m; m.parent()
[  0   2]
[  0 2/5]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

m = matrix({(0,1): 2, (1,1): 2/5}); m; m.parent()

sage: m = matrix(QQ, 2, 3, {(1,1): 2}); m; m.parent()
[0 0 0]
[0 2 0]
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

>>> from sage.all import *
>>> m = matrix(QQ, Integer(2), Integer(3), {(Integer(1),Integer(1)): Integer(2)}); m; m.parent()
[0 0 0]
[0 2 0]
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

m = matrix(QQ, 2, 3, {(1,1): 2}); m; m.parent()

sage: import numpy                                                              # needs numpy
sage: n = numpy.array([[1,2], [3,4]], float)                                    # needs numpy
sage: m = matrix(n); m; m.parent()                                              # needs numpy
[1.0 2.0]
[3.0 4.0]
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

>>> from sage.all import *
>>> import numpy                                                              # needs numpy
>>> n = numpy.array([[Integer(1),Integer(2)], [Integer(3),Integer(4)]], float)                                    # needs numpy
>>> m = matrix(n); m; m.parent()                                              # needs numpy
[1.0 2.0]
[3.0 4.0]
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

import numpy                                                              # needs numpy
n = numpy.array([[1,2], [3,4]], float)                                    # needs numpy
m = matrix(n); m; m.parent()                                              # needs numpy

sage: v = vector(ZZ, [1, 10, 100])
sage: m = matrix(v); m; m.parent()
[  1  10 100]
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
sage: m = matrix(GF(7), v); m; m.parent()
[1 3 2]
Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7
sage: m = matrix(GF(7), 3, 1, v); m; m.parent()
[1]
[3]
[2]
Full MatrixSpace of 3 by 1 dense matrices over Finite Field of size 7

>>> from sage.all import *
>>> v = vector(ZZ, [Integer(1), Integer(10), Integer(100)])
>>> m = matrix(v); m; m.parent()
[  1  10 100]
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
>>> m = matrix(GF(Integer(7)), v); m; m.parent()
[1 3 2]
Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7
>>> m = matrix(GF(Integer(7)), Integer(3), Integer(1), v); m; m.parent()
[1]
[3]
[2]
Full MatrixSpace of 3 by 1 dense matrices over Finite Field of size 7

v = vector(ZZ, [1, 10, 100])
m = matrix(v); m; m.parent()
m = matrix(GF(7), v); m; m.parent()
m = matrix(GF(7), 3, 1, v); m; m.parent()

sage: matrix(pari.mathilbert(3))                                                # needs sage.libs.pari
[  1 1/2 1/3]
[1/2 1/3 1/4]
[1/3 1/4 1/5]

>>> from sage.all import *
>>> matrix(pari.mathilbert(Integer(3)))                                                # needs sage.libs.pari
[  1 1/2 1/3]
[1/2 1/3 1/4]
[1/3 1/4 1/5]

matrix(pari.mathilbert(3))                                                # needs sage.libs.pari

sage: g = graphs.PetersenGraph()                                                # needs sage.graphs
sage: m = matrix(g); m; m.parent()                                              # needs sage.graphs
[0 1 0 0 1 1 0 0 0 0]
[1 0 1 0 0 0 1 0 0 0]
[0 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 0 0 0 1 0]
[1 0 0 1 0 0 0 0 0 1]
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 1 0 0 0 1]
[0 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 0]
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring

>>> from sage.all import *
>>> g = graphs.PetersenGraph()                                                # needs sage.graphs
>>> m = matrix(g); m; m.parent()                                              # needs sage.graphs
[0 1 0 0 1 1 0 0 0 0]
[1 0 1 0 0 0 1 0 0 0]
[0 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 0 0 0 1 0]
[1 0 0 1 0 0 0 0 0 1]
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 1 0 0 0 1]
[0 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 0]
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring

g = graphs.PetersenGraph()                                                # needs sage.graphs
m = matrix(g); m; m.parent()                                              # needs sage.graphs

sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent()
Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring

>>> from sage.all import *
>>> matrix(ZZ, Integer(10), Integer(10), range(Integer(100)), sparse=True).parent()
Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring

matrix(ZZ, 10, 10, range(100), sparse=True).parent()

sage: R = PolynomialRing(QQ, 9, 'x')
sage: A = matrix(R, 3, 3, R.gens()); A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
sage: det(A)
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(9), 'x')
>>> A = matrix(R, Integer(3), Integer(3), R.gens()); A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
>>> det(A)
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

R = PolynomialRing(QQ, 9, 'x')
A = matrix(R, 3, 3, R.gens()); A
det(A)

sage: M = Matrix([[1,2,3], [4,5,6], [7,8,9]], immutable=True)
sage: M[0] = [9,9,9]
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).

>>> from sage.all import *
>>> M = Matrix([[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)], [Integer(7),Integer(8),Integer(9)]], immutable=True)
>>> M[Integer(0)] = [Integer(9),Integer(9),Integer(9)]
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).

M = Matrix([[1,2,3], [4,5,6], [7,8,9]], immutable=True)
M[0] = [9,9,9]

Using row_keys and column_keys:

sage: M = matrix([[1,2,3], [4,5,6]],
....:            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
  From: Free module generated by {'a', 'b', 'c'} over Integer Ring
  To:   Free module generated by {'u', 'v'} over Integer Ring
sage: print(M._unicode_art_matrix())
  a b c
u⎛1 2 3⎞
v⎝4 5 6⎠

>>> from sage.all import *
>>> M = matrix([[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)]],
...            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
  From: Free module generated by {'a', 'b', 'c'} over Integer Ring
  To:   Free module generated by {'u', 'v'} over Integer Ring
>>> print(M._unicode_art_matrix())
  a b c
u⎛1 2 3⎞
v⎝4 5 6⎠

M = matrix([[1,2,3], [4,5,6]],
           column_keys=['a','b','c'], row_keys=['u','v']); M
print(M._unicode_art_matrix())

It is allowed to specify dimensions redundantly:

sage: M = matrix(2, 3, [[1,2,3], [4,5,6]],
....:            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
From: Free module generated by {'a', 'b', 'c'} over Integer Ring
To:   Free module generated by {'u', 'v'} over Integer Ring

>>> from sage.all import *
>>> M = matrix(Integer(2), Integer(3), [[Integer(1),Integer(2),Integer(3)], [Integer(4),Integer(5),Integer(6)]],
...            column_keys=['a','b','c'], row_keys=['u','v']); M
Generic morphism:
From: Free module generated by {'a', 'b', 'c'} over Integer Ring
To:   Free module generated by {'u', 'v'} over Integer Ring

M = matrix(2, 3, [[1,2,3], [4,5,6]],
           column_keys=['a','b','c'], row_keys=['u','v']); M

AUTHORS:

  • William Stein: Initial implementation

  • Jason Grout (2008-03): almost a complete rewrite, with bits and pieces from the original implementation

  • Jeroen Demeyer (2016-02-05): major clean up, see Issue #20015 and Issue #20016

  • Jeroen Demeyer (2018-02-20): completely rewritten using MatrixArgs, see Issue #24742

sage.matrix.constructor.options(*get_value, **set_value)[source]

Global options for matrices.

OPTIONS:

  • format_numeric – (default: {:.{prec}}) string used for formatting floating point numbers of an (optional) precision prec; only supported for entry types implementing __format__

  • max_cols – (default: 49) maximum number of columns to display

  • max_rows – (default: 19) maximum number of rows to display

  • precision – (default: None) number of digits to display for floating point entries; if None, the exact representation is used instead. This option is also set by the IPython magic %precision.

    EXAMPLES:

    sage: matrix.options.max_cols = 6
sage: matrix.options.max_rows = 3
sage: matrix(ZZ, 3, 6)
[0 0 0 0 0 0]
[0 0 0 0 0 0]
[0 0 0 0 0 0]
sage: matrix(ZZ, 3, 7)
3 x 7 dense matrix over Integer Ring...
sage: matrix(ZZ, 4, 6)
4 x 6 dense matrix over Integer Ring...
sage: matrix.options._reset()

    >>> from sage.all import *
>>> matrix.options.max_cols = Integer(6)
>>> matrix.options.max_rows = Integer(3)
>>> matrix(ZZ, Integer(3), Integer(6))
[0 0 0 0 0 0]
[0 0 0 0 0 0]
[0 0 0 0 0 0]
>>> matrix(ZZ, Integer(3), Integer(7))
3 x 7 dense matrix over Integer Ring...
>>> matrix(ZZ, Integer(4), Integer(6))
4 x 6 dense matrix over Integer Ring...
>>> matrix.options._reset()

    matrix.options.max_cols = 6
matrix.options.max_rows = 3
matrix(ZZ, 3, 6)
matrix(ZZ, 3, 7)
matrix(ZZ, 4, 6)
matrix.options._reset()

    The precision can also be set via the IPython magic:

    sage: from sage.repl.interpreter import get_test_shell
sage: shell = get_test_shell()
sage: shell.run_cell('%precision 5')
'%.5f'
sage: matrix.options.precision
5
sage: A = matrix(RR, [[200/3]]); A
[66.667]

    >>> from sage.all import *
>>> from sage.repl.interpreter import get_test_shell
>>> shell = get_test_shell()
>>> shell.run_cell('%precision 5')
'%.5f'
>>> matrix.options.precision
5
>>> A = matrix(RR, [[Integer(200)/Integer(3)]]); A
[66.667]

    from sage.repl.interpreter import get_test_shell
shell = get_test_shell()
shell.run_cell('%precision 5')
matrix.options.precision
A = matrix(RR, [[200/3]]); A

    The number format can be specified as well:

    sage: matrix.options.format_numeric = '{:.{prec}e}'
sage: A
[6.66667e+1]
sage: matrix.options.format_numeric = '{:.{prec}f}'
sage: A
[66.66667]
sage: matrix.options.format_numeric = '{:+.{prec}g}'
sage: A
[+66.667]
sage: matrix.options._reset()

    >>> from sage.all import *
>>> matrix.options.format_numeric = '{:.{prec}e}'
>>> A
[6.66667e+1]
>>> matrix.options.format_numeric = '{:.{prec}f}'
>>> A
[66.66667]
>>> matrix.options.format_numeric = '{:+.{prec}g}'
>>> A
[+66.667]
>>> matrix.options._reset()

    matrix.options.format_numeric = '{:.{prec}e}'
A
matrix.options.format_numeric = '{:.{prec}f}'
A
matrix.options.format_numeric = '{:+.{prec}g}'
A
matrix.options._reset()

See GlobalOptions for more features of these options.