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 Check that the determinant is computed from a cached charpoly properly: sage: A = matrix(RR, [ [1, 0, 1/2], ... [0, 1, 0 ], ... [0, 0, -2 ] ]) sage: B = copy(A) sage: _ = A.charpoly() sage: A.determinant() == B.determinant() True AUTHORS: •Unknown: No author specified in the file from 2009-06-25 •Sebastian Pancratz (2009-06-25): Use the division-free algorithm for charpoly •Thierry Monteil (2010-10-05): Bugfix for trac ticket #10063, so that the determinant is computed even for rings for which the is_field method is not implemented. diagonal() Return the diagonal entries of self. OUTPUT: A list containing the entries of the matrix that have equal row and column indices, in order of the indices. Behavior is not limited to square matrices. EXAMPLES: sage: A = matrix([[2,5],[3,7]]); A [2 5] [3 7] sage: A.diagonal() [2, 7] Two rectangular matrices. sage: B = matrix(3, 7, range(21)); B [ 0 1 2 3 4 5 6] [ 7 8 9 10 11 12 13] [14 15 16 17 18 19 20] sage: B.diagonal() [0, 8, 16] sage: C = matrix(3, 2, range(6)); C [0 1] [2 3] [4 5] sage: C.diagonal() [0, 3] Empty matrices behave properly. sage: E = matrix(0, 5, []); E [] sage: E.diagonal() [] echelon_form(algorithm=’default’, cutoff=0, **kwds) Return the echelon form of self. 152 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 Note: This row reduction does not use division if the matrix is not over a field (e.g., if the matrix is over the integers). If you want to calculate the echelon form using division, then use rref(), which assumes that the matrix entries are in a field (specifically, the field of fractions of the base ring of the matrix). INPUT: •algorithm – string. Which algorithm to use. Choices are –’default’: Let Sage choose an algorithm (default). –’classical’: Gauss elimination. –’strassen’: use a Strassen divide and conquer algorithm (if available) •cutoff – integer. Only used if the Strassen algorithm is selected. •transformation – boolean. Whether to also return the transformation matrix. Some matrix backends do not provide this information, in which case this option is ignored. OUTPUT: The reduced row echelon form of self, as an immutable matrix. Note that self is not changed by this command. Use echelonize() to change self in place. If the optional parameter transformation=True is specified, the output consists of a pair (E, T ) of matrices where E is the echelon form of self and T is the transformation matrix. EXAMPLES: sage: MS = MatrixSpace(GF(19),2,3) sage: C = MS.matrix([1,2,3,4,5,6]) sage: C.rank() 2 sage: C.nullity() 0 sage: C.echelon_form() [ 1 0 18] [ 0 1 2] The matrix library used for Z/p-matrices does not return the transformation matrix, transformation option is ignored: so the sage: C.echelon_form(transformation=True) [ 1 0 18] [ 0 1 2] sage: D = matrix(ZZ, 2, 3, [1,2,3,4,5,6]) sage: D.echelon_form(transformation=True) ( [1 2 3] [ 1 0] [0 3 6], [ 4 -1] ) sage: E, T = D.echelon_form(transformation=True) sage: T*D == E True echelonize(algorithm=’default’, cutoff=0, **kwds) Transform self into a matrix in echelon form over the same base ring as self. Note: This row reduction does not use division if the matrix is not over a field (e.g., if the matrix is over the integers). If you want to calculate the echelon form using division, then use rref(), which assumes 153
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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