Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 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 = numpy.complex(0,1) sage: entries = numpy.array([2.0+j, 8.1, 3.4+2.6*j]); entries array([ 2.0+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: Cannot convert non-integral float to integer 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 having a length. sage: diagonal_matrix(QQ, 5, 10) Traceback (most recent call last): ... TypeError: unable to determine number of entries for diagonal matrix construction 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) 30 Chapter 2. Matrix Constructor
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 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 are limited, even though they may have a length. sage: diagonal_matrix(x^2) Traceback (most recent call last): ... TypeError: diagonal matrix entries are not a supported type (list, tuple, vector, or NumPy array AUTHOR: •Rob Beezer (2011-01-11): total rewrite sage.matrix.constructor.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. 31
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