Sharedsupport / 2019-04-11-163506-matrix.sagewsOpen in CoCalc
Author: William A. Stein
License: GNU General Public License v3.0
Description: Examples for support purposes.
block_matrix?
File: /ext/sage/sage-8.6_1804/local/lib/python2.7/site-packages/sage/matrix/special.py Signature : block_matrix(*args, **kwds) Docstring : 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 Historically, giving only a flat list of submatrices, whose number was a perfect square, would create a new matrix by laying out the submatrices in a square grid. This behavior is now deprecated. sage: A = matrix(2, 3, range(6)) sage: B = matrix(3, 3, range(9)) sage: block_matrix([A, A, B, B]) doctest:...: DeprecationWarning: invocation of block_matrix with just a list whose length is a perfect square is deprecated. See the documentation for details. [0 1 2|0 1 2] [3 4 5|3 4 5] [-----+-----] [0 1 2|0 1 2] [3 4 5|3 4 5] [6 7 8|6 7 8] Historically, a flat list of matrices whose number is not a perfect square, with no specification of the number of rows or columns, would raise an error. This behavior continues, but could be removed when the deprecation above is completed. sage: A = matrix(2, 3, range(6)) sage: B = matrix(3, 3, range(9)) sage: block_matrix([A, A, A, B, B, B]) Traceback (most recent call last): ... ValueError: must specify nrows or ncols for non-square block matrix.