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: n= m^(-1); n [ 0.333333333333 + 0.333333333333*I 0.166666666667 - 0.166666666667*I] [ -0.166666666667 - 0.333333333333*I 0.0833333333333 + 0.0833333333333*I] To compute eigenvalues the use the functions left_eigenvectors or right_eigenvectors sage: p,e = m.right_eigenvectors() the result of eigen is a pair (p,e), where p is a list of eigenvalues and the e is a matrix whose columns are the eigenvectors. To solve a linear system Ax = b where A = [[1,2] and b = [5,6] [3,4]] sage: b = vector(CDF,[5,6]) sage: m.solve_right(b) (2.66666666667 + 0.666666666667*I, -0.333333333333 - 1.16666666667*I) See the commands qr, lu, and svd for QR, LU, and singular value decomposition. 398 Chapter 21. Dense matrices over the Complex Double Field using NumPy
CHAPTER TWENTYTWO DENSE MATRICES OVER MULTIVARIATE POLYNOMIALS OVER FIELDS Dense matrices over multivariate polynomials over fields This implementation inherits from Matrix_generic_dense, i.e. it is not optimized for speed only some methods were added. AUTHOR: • Martin Albrecht class sage.matrix.matrix_mpolynomial_dense.Matrix_mpolynomial_dense Bases: sage.matrix.matrix_generic_dense.Matrix_generic_dense Dense matrix over a multivariate polynomial ring over a field. determinant(algorithm=None) Return the determinant of this matrix INPUT: •algorithm – ignored EXAMPLES: We compute the determinant of the arbitrary 3x3 matrix: sage: R = PolynomialRing(QQ, 9, ’x’) sage: A = matrix(R, 3, R.gens()) sage: A [x0 x1 x2] [x3 x4 x5] [x6 x7 x8] sage: A.determinant() -x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8 We check if two implementations agree on the result: sage: R. = QQ[] sage: C = random_matrix(R, 2, 2, terms=2) sage: C [-6/5*x*y - y^2 -6*y^2 - 1/4*y] [ -1/3*x*y - 3 x*y - x] sage: C.determinant() 399
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Sage Reference Manual: Matrices and
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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