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 True right_kernel(*args, **kwds) Returns the right kernel of this matrix, as a vector space or free module. This is the set of vectors x such that self*x = 0. Note: For the left kernel, use left_kernel(). The method kernel() is exactly equal to left_kernel(). INPUT: •algorithm - default: ‘default’ - a keyword that selects the algorithm employed. Allowable values are: –‘default’ - allows the algorithm to be chosen automatically –‘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 used to construct the left kernel. 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. A vector space or free module whose degree equals the number of columns in self and contains all the vectors x such that self*x = 0. If self has 0 columns, the kernel has dimension 0, while if self has 0 rows the kernel is the entire ambient vector space. The result is cached. Requesting the right kernel a second time, but with a different basis format, will return the cached result with the format from the first computation. Note: For more detailed documentation on the selection of algorithms used and a more flexible method for computing a basis matrix for a right kernel (rather than computing a vector space), see right_kernel_matrix(), which powers the computations for this method. EXAMPLES: sage: A = matrix(QQ, [[0, 0, 1, 2, 2, -5, 3], ... [-1, 5, 2, 2, 1, -7, 5], ... [0, 0, -2, -3, -3, 8, -5], ... [-1, 5, 0, -1, -2, 1, 0]]) sage: K = A.right_kernel(); K Vector space of degree 7 and dimension 4 over Rational Field Basis matrix: [ 1 0 0 0 -1 -1 -1] [ 0 1 0 0 5 5 5] 236 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 0 0 1 0 -1 -2 -3] [ 0 0 0 1 0 1 1] sage: A*K.basis_matrix().transpose() == zero_matrix(QQ, 4, 4) True The default is basis vectors that form a matrix in echelon form. A “pivot basis” instead has a basis matrix where the columns of an identity matrix are in the locations of the non-pivot columns of the original matrix. This alternate format is available whenever the base ring is a field. sage: A = matrix(QQ, [[0, 0, 1, 2, 2, -5, 3], ... [-1, 5, 2, 2, 1, -7, 5], ... [0, 0, -2, -3, -3, 8, -5], ... [-1, 5, 0, -1, -2, 1, 0]]) sage: A.rref() [ 1 -5 0 0 1 1 -1] [ 0 0 1 0 0 -1 1] [ 0 0 0 1 1 -2 1] [ 0 0 0 0 0 0 0] sage: A.nonpivots() (1, 4, 5, 6) sage: K = A.right_kernel(basis=’pivot’); K Vector space of degree 7 and dimension 4 over Rational Field User basis matrix: [ 5 1 0 0 0 0 0] [-1 0 0 -1 1 0 0] [-1 0 1 2 0 1 0] [ 1 0 -1 -1 0 0 1] sage: A*K.basis_matrix().transpose() == zero_matrix(QQ, 4, 4) True Matrices may have any field as a base ring. Number fields are computed by PARI library code, matrices over GF (2) are computed by the M4RI library, and matrices over the rationals are computed by the IML library. For any of these specialized cases, general-purpose code can be called instead with the keyword setting algorithm=’generic’. Over an arbitrary field, with two basis formats. Same vector space, different bases. sage: F. = FiniteField(5^2) sage: A = matrix(F, 3, 4, [[ 1, a, 1+a, a^3+a^5], ... [ a, a^4, a+a^4, a^4+a^8], ... [a^2, a^6, a^2+a^6, a^5+a^10]]) sage: K = A.right_kernel(); K Vector space of degree 4 and dimension 2 over Finite Field in a of size 5^2 Basis matrix: [ 1 0 3*a + 4 2*a + 2] [ 0 1 2*a 3*a + 3] sage: A*K.basis_matrix().transpose() == zero_matrix(F, 3, 2) True sage: B = copy(A) sage: P = B.right_kernel(basis = ’pivot’); P Vector space of degree 4 and dimension 2 over Finite Field in a of size 5^2 User basis matrix: [ 4 4 1 0] [ a + 2 3*a + 3 0 1] sage: B*P.basis_matrix().transpose() == zero_matrix(F, 3, 2) True sage: K == P True 237
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