General matrix Constructor and display options#
- sage.matrix.constructor.Matrix(*args, **kwds)#
Create a matrix.
This implements the
matrixconstructor:sage: matrix([[1,2],[3,4]]) [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]
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) are optional.
Positional and keyword arguments:
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 toZZif no entries are given.nrows– the number of rows in the matrix.ncols– the number of columns in the matrix.entries– see examples below.
If either
nrowsorncolsis given as keyword argument, then no positional argumentsnrowsandncolsmay be given.Keyword-only arguments:
sparse– (boolean) create a sparse matrix. This defaults toTruewhen the entries are given as a dictionary, otherwise defaults toFalse.space– matrix space which will be the parent of the output matrix. This determinesring,nrows,ncolsandsparse.immutable– (boolean) make the matrix immutable. By default, the output matrix is mutable.
OUTPUT:
a matrix
EXAMPLES:
sage: m = matrix(2); m; m.parent() [0 0] [0 0] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
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
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
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]
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]
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
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
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
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
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
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
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
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]
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
sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent() Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring
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
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).
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 github issue #20015 and github issue #20016
Jeroen Demeyer (2018-02-20): completely rewritten using
MatrixArgs, see github issue #24742
- sage.matrix.constructor.matrix(*args, **kwds)#
Create a matrix.
This implements the
matrixconstructor:sage: matrix([[1,2],[3,4]]) [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]
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) are optional.
Positional and keyword arguments:
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 toZZif no entries are given.nrows– the number of rows in the matrix.ncols– the number of columns in the matrix.entries– see examples below.
If either
nrowsorncolsis given as keyword argument, then no positional argumentsnrowsandncolsmay be given.Keyword-only arguments:
sparse– (boolean) create a sparse matrix. This defaults toTruewhen the entries are given as a dictionary, otherwise defaults toFalse.space– matrix space which will be the parent of the output matrix. This determinesring,nrows,ncolsandsparse.immutable– (boolean) make the matrix immutable. By default, the output matrix is mutable.
OUTPUT:
a matrix
EXAMPLES:
sage: m = matrix(2); m; m.parent() [0 0] [0 0] Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
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
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
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]
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]
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
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
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
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
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
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
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
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]
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
sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent() Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring
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
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).
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 github issue #20015 and github issue #20016
Jeroen Demeyer (2018-02-20): completely rewritten using
MatrixArgs, see github issue #24742
- sage.matrix.constructor.options(*get_value, **set_value)#
Global options for matrices.
OPTIONS:
format_numeric– (default:{:.{prec}}) string used for formatting floating point numbers of an (optional) precisionprec; only supported for entry types implementing__format__max_cols– (default:49) maximum number of columns to displaymax_rows– (default:19) maximum number of rows to displayprecision– (default:None) number of digits to display for floating point entries; ifNone, 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()
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]
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()
See
GlobalOptionsfor more features of these options.