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 kernel. sage: A = matrix(QQ, 2, 0) sage: A.right_kernel() Vector space of degree 0 and dimension 0 over Rational Field Basis matrix: [] sage: A = matrix(QQ, 0, 2) sage: A.right_kernel() Vector space of degree 2 and dimension 2 over Rational Field Basis matrix: [1 0] [0 1] Every vector is in the kernel of a zero matrix, the dimension is the number of columns. sage: A = zero_matrix(QQ, 10, 20) sage: A.right_kernel() Vector space of degree 20 and dimension 20 over Rational Field Basis matrix: 20 x 20 dense matrix over Rational Field Results are cached as the right kernel of the matrix. Subsequent requests for the right kernel will return the cached result, without regard for new values of the algorithm or format keyword. Work with a copy if you need a new right kernel, or perhaps investigate the right_kernel_matrix() method, which does not cache its results and is more flexible. sage: A = matrix(QQ, 3, range(9)) sage: K1 = A.right_kernel(basis=’echelon’) sage: K1 Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 -2 1] sage: K2 = A.right_kernel(basis=’pivot’) sage: K2 Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 -2 1] sage: K1 is K2 True sage: B = copy(A) sage: K3 = B.kernel(basis=’pivot’) sage: K3 Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1] sage: K3 is K1 False sage: K3 == K1 True right_kernel_matrix(*args, **kwds) Returns a matrix whose rows form a basis for the right kernel of self. INPUT: •algorithm - default: ‘default’ - a keyword that selects the algorithm employed. Allowable values are: –‘default’ - allows the algorithm to be chosen automatically 240 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 –‘generic’ - naive algorithm usable for matrices over any field –‘pari’ - PARI library code for matrices over number fields or the integers –‘padic’ - padic algorithm from IML library for matrices over the rationals and integers –‘pluq’ - PLUQ matrix factorization for matrices mod 2 •basis - default: ‘echelon’ - a keyword that describes the format of the basis returned. Allowable values are: OUTPUT: –‘echelon’: the basis matrix is in echelon form –‘pivot’ : each basis vector is computed from the reduced row-echelon form of self by placing a single one in a non-pivot column and zeros in the remaining non-pivot columns. Only available for matrices over fields. –‘LLL’: an LLL-reduced basis. Only available for matrices over the integers. –‘computed’: no work is done to transform the basis, it is returned exactly as provided by whichever routine actually computed the basis. Request this for the least possible computation possible, but with no guarantees about the format of the basis. A matrix X whose rows are an independent set spanning the right kernel of self. self*X.transpose() is a zero matrix. The output varies depending on the choice of algorithm and the format chosen by basis. The results of this routine are not cached, so you can call it again with different options to get possibly different output (like the basis format). Conversely, repeated calls on the same matrix will always start from scratch. Note: If you want to get the most basic description of a kernel, with a minimum of overhead, then ask for the right kernel matrix with the basis format requested as ‘computed’. You are then free to work with the output for whatever purpose. For a left kernel, call this method on the transpose of your matrix. For greater convenience, plus cached results, request an actual vector space or free module with right_kernel() or left_kernel(). EXAMPLES: Over the Rational Numbers: Kernels are computed by the IML library in _right_kernel_matrix(). Setting the algorithm keyword to ‘default’, ‘padic’ or unspecified will yield the same result, as there is no optional behavior. The ‘computed’ format of the basis vectors are exactly the negatives of the vectors in the ‘pivot’ format. sage: A = matrix(QQ, [[1, 0, 1, -3, 1], ... [-5, 1, 0, 7, -3], ... [0, -1, -4, 6, -2], ... [4, -1, 0, -6, 2]]) sage: C = A.right_kernel_matrix(algorithm=’default’, basis=’computed’); C [-1 2 -2 -1 0] [ 1 2 0 0 -1] sage: A*C.transpose() == zero_matrix(QQ, 4, 2) True sage: P = A.right_kernel_matrix(algorithm=’padic’, basis=’pivot’); P [ 1 -2 2 1 0] [-1 -2 0 0 1] sage: A*P.transpose() == zero_matrix(QQ, 4, 2) So 241
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