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: A.right_kernel_matrix().parent() Full MatrixSpace of 0 by 0 dense matrices over Finite Field of size 7 TESTS: The “usual” quaternions are a non-commutative ring and computations of kernels over these rings are not implemented. sage: Q. = QuaternionAlgebra(-1,-1) sage: A = matrix(Q, 2, [i,j,-1,k]) sage: A.right_kernel_matrix() Traceback (most recent call last): ... NotImplementedError: Cannot compute a matrix kernel over Quaternion Algebra (-1, -1) with ba We test error messages for improper choices of the ‘algorithm’ keyword. sage: matrix(ZZ, 2, 2).right_kernel_matrix(algorithm=’junk’) Traceback (most recent call last): ... ValueError: matrix kernel algorithm ’junk’ not recognized sage: matrix(GF(2), 2, 2).right_kernel_matrix(algorithm=’padic’) Traceback (most recent call last): ... ValueError: ’padic’ matrix kernel algorithm only available over the rationals and the intege sage: matrix(QQ, 2, 2).right_kernel_matrix(algorithm=’pari’) Traceback (most recent call last): ... ValueError: ’pari’ matrix kernel algorithm only available over non-trivial number fields and sage: matrix(Integers(6), 2, 2).right_kernel_matrix(algorithm=’generic’) Traceback (most recent call last): ... ValueError: ’generic’ matrix kernel algorithm only available over a field, not over Ring of sage: matrix(QQ, 2, 2).right_kernel_matrix(algorithm=’pluq’) Traceback (most recent call last): ... ValueError: ’pluq’ matrix kernel algorithm only available over integers mod 2, not over Rati We test error messages for improper basis format requests. sage: matrix(ZZ, 2, 2).right_kernel_matrix(basis=’junk’) Traceback (most recent call last): ... ValueError: matrix kernel basis format ’junk’ not recognized sage: matrix(ZZ, 2, 2).right_kernel_matrix(basis=’pivot’) Traceback (most recent call last): ... ValueError: pivot basis only available over a field, not over Integer Ring sage: matrix(QQ, 2, 2).right_kernel_matrix(basis=’LLL’) Traceback (most recent call last): ... ValueError: LLL-reduced basis only available over the integers, not over Rational Field Finally, error messages for the ‘proof’ keyword. sage: matrix(ZZ, 2, 2).right_kernel_matrix(proof=’junk’) Traceback (most recent call last): ... ValueError: ’proof’ must be one of True, False or None, not junk sage: matrix(QQ, 2, 2).right_kernel_matrix(proof=True) 248 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 Traceback (most recent call last): ... ValueError: ’proof’ flag only valid for matrices over the integers AUTHOR: •Rob Beezer (2011-02-05) right_nullity() Return the right nullity of this matrix, which is the dimension of the right kernel. EXAMPLES: sage: A = MatrixSpace(QQ,3,2)(range(6)) sage: A.right_nullity() 0 sage: A = matrix(ZZ,3,range(9)) sage: A.right_nullity() 1 rook_vector(check=False) Return the rook vector of the matrix self. Let A be an m by n (0,1)-matrix with m ≤ n. We identify A with a chessboard where rooks can be placed on the fields (i, j) with a i,j = 1. The number r k = p k (A) (the permanental k-minor) counts the number of ways to place k rooks on this board so that no rook can attack another. The rook vector of the matrix A is the list consisting of r 0 , r 1 , . . . , r m . The rook polynomial is defined by r(x) = ∑ m k=0 r kx k . INPUT: •self – an m by n (0,1)-matrix with m ≤ n •check – Boolean (default: False) determining whether to check that self is a (0,1)-matrix. OUTPUT: The rook vector of the matrix self. EXAMPLES: sage: M = MatrixSpace(ZZ,3,6) sage: A = M([1,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,1]) sage: A.rook_vector() [1, 12, 40, 36] sage: R. = PolynomialRing(ZZ) sage: rv = A.rook_vector() sage: rook_polynomial = sum([rv[k] * x^k for k in range(len(rv))]) sage: rook_polynomial 36*x^3 + 40*x^2 + 12*x + 1 AUTHORS: •Jaap Spies (2006-02-24) row_module(base_ring=None) Return the free module over the base ring spanned by the rows of self. EXAMPLES: 249
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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