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 A small matrix that does not fit into any of the usual categories of normal matrices. sage: A = matrix(ZZ, [[1, -1], ... [1, 1]]) sage: A.is_normal() True sage: not A.is_hermitian() and not A.is_skew_symmetric() True Sage has several fields besides the entire complex numbers where conjugation is non-trivial. sage: F. = QuadraticField(-7) sage: C = matrix(F, [[-2*b - 3, 7*b - 6, -b + 3], ... [-2*b - 3, -3*b + 2, -2*b], ... [ b + 1, 0, -2]]) sage: C = C*C.conjugate_transpose() sage: C.is_normal() True A matrix that is nearly normal, but for a non-real diagonal entry. sage: A = matrix(QQbar, [[ 2, 2-I, 1+4*I], ... [ 2+I, 3+I, 2-6*I], ... [1-4*I, 2+6*I, 5]]) sage: A.is_normal() False sage: A[1,1] = 132 sage: A.is_normal() True Rectangular matrices are never normal. sage: A = matrix(QQbar, 3, 4) sage: A.is_normal() False A square, empty matrix is trivially normal. sage: A = matrix(QQ, 0, 0) sage: A.is_normal() True AUTHOR: •Rob Beezer (2011-03-31) is_one() Return True if this matrix is the identity matrix. EXAMPLES: sage: m = matrix(QQ,2,range(4)) sage: m.is_one() False sage: m = matrix(QQ,2,[5,0,0,5]) sage: m.is_one() False sage: m = matrix(QQ,2,[1,0,0,1]) sage: m.is_one() True sage: m = matrix(QQ,2,[1,1,1,1]) 188 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: m.is_one() False is_positive_definite() Determines if a real or symmetric matrix is positive definite. A square matrix A is postive definite if it is symmetric with real entries or Hermitan with complex entries, and for every non-zero vector ⃗x ⃗x ∗ A⃗x > 0 Here ⃗x ∗ is the conjugate-transpose, which can be simplified to just the transpose in the real case. ALGORITHM: A matrix is positive definite if and only if the diagonal entries from the indefinite factorization are all positive (see indefinite_factorization()). So this algorithm is of order n^3/3 and may be applied to matrices with elements of any ring that has a fraction field contained within the reals or complexes. INPUT: Any square matrix. OUTPUT: This routine will return True if the matrix is square, symmetric or Hermitian, and meets the condition above for the quadratic form. The base ring for the elements of the matrix needs to have a fraction field implemented and the computations that result from the indefinite factorization must be convertable to real numbers that are comparable to zero. EXAMPLES: A real symmetric matrix that is positive definite, as evidenced by the positive entries for the diagonal matrix of the indefinite factorization and the postive determinants of the leading principal submatrices. sage: A = matrix(QQ, [[ 4, -2, 4, 2], ... [-2, 10, -2, -7], ... [ 4, -2, 8, 4], ... [ 2, -7, 4, 7]]) sage: A.is_positive_definite() True sage: _, d = A.indefinite_factorization(algorithm=’symmetric’) sage: d (4, 9, 4, 1) sage: [A[:i,:i].determinant() for i in range(1,A.nrows()+1)] [4, 36, 144, 144] A real symmetric matrix which is not positive definite, along with a vector that makes the quadratic form negative. sage: A = matrix(QQ, [[ 3, -6, 9, 6, -9], ... [-6, 11, -16, -11, 17], ... [ 9, -16, 28, 16, -40], ... [ 6, -11, 16, 9, -19], ... [-9, 17, -40, -19, 68]]) sage: A.is_positive_definite() False sage: _, d = A.indefinite_factorization(algorithm=’symmetric’) sage: d (3, -1, 5, -2, -1) 189
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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