Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[3, -3, -15, 30, -30]
sage: A.is_positive_definite()
False
A matrix over CDF that is not positive definite.
sage: B = matrix(CDF, [[ 2, 4 - 2*I, 2 + 2*I],
... [4 + 2*I, 8, 10*I],
... [2 - 2*I, -10*I, -3]])
sage: B.is_hermitian()
True
sage: [ev.real() for ev in B.eigenvalues()]
[15.88..., 0.08..., -8.97...]
sage: [round(B[:i,:i].determinant().real()) for i in range(1, B.nrows()+1)]
[2, -4, -12]
sage: B.is_positive_definite()
False
A large random matrix that is guaranteed by theory to be positive definite.
sage: R = random_matrix(CDF, 200)
sage: H = R.conjugate_transpose()*R
sage: H.is_positive_definite()
True
TESTS:
A trivially small case.
sage: S = matrix(CDF, [])
sage: S.nrows(), S.ncols()
(0, 0)
sage: S.is_positive_definite()
True
A rectangular matrix will never be positive definite.
sage: R = matrix(RDF, 2, 3, range(6))
sage: R.is_positive_definite()
False
A non-Hermitian matrix will never be positive definite.
sage: T = matrix(CDF, 8, 8, range(64))
sage: T.is_positive_definite()
False
AUTHOR:
•Rob Beezer (2012-05-28)
is_symmetric(tol=1e-12)
Return whether this matrix is symmetric, to the given tolerance.
EXAMPLES:
sage: m = matrix(RDF,2,2,range(4)); m
[0.0 1.0]
[2.0 3.0]
sage: m.is_symmetric()
False
sage: m[1,0] = 1.1; m
373