Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
INPUT:
Any matrix over RDF or CDF.
OUTPUT:
True if and only if the matrix is square, Hermitian, and meets the condition above on the quadratic form.
The result is cached.
IMPLEMENTATION:
The existence of a Cholesky decomposition and the positive definite property are equivalent. So this
method and the cholesky() method compute and cache both the Cholesky decomposition and the
positive-definiteness. So the is_positive_definite() method or catching a ValueError from
the cholesky() method are equally expensive computationally and if the decomposition exists, it is
cached as a side-effect of either routine.
EXAMPLES:
A matrix over RDF that is positive definite.
sage: M = matrix(RDF,[[ 1, 1, 1, 1, 1],
... [ 1, 5, 31, 121, 341],
... [ 1, 31, 341, 1555, 4681],
... [ 1,121, 1555, 7381, 22621],
... [ 1,341, 4681, 22621, 69905]])
sage: M.is_symmetric()
True
sage: M.eigenvalues()
[77547.66..., 82.44..., 2.41..., 0.46..., 0.011...]
sage: [round(M[:i,:i].determinant()) for i in range(1, M.nrows()+1)]
[1, 4, 460, 27936, 82944]
sage: M.is_positive_definite()
True
A matrix over CDF that is positive definite.
sage: C = matrix(CDF, [[ 23, 17*I + 3, 24*I + 25, 21*I],
... [ -17*I + 3, 38, -69*I + 89, 7*I + 15],
... [-24*I + 25, 69*I + 89, 976, 24*I + 6],
... [ -21*I, -7*I + 15, -24*I + 6, 28]])
sage: C.is_hermitian()
True
sage: [x.real() for x in C.eigenvalues()]
[991.46..., 55.96..., 3.69..., 13.87...]
sage: [round(C[:i,:i].determinant().real()) for i in range(1, C.nrows()+1)]
[23, 576, 359540, 2842600]
sage: C.is_positive_definite()
True
A matrix over RDF that is not positive definite.
sage: A = matrix(RDF, [[ 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_symmetric()
True
sage: A.eigenvalues()
[108.07..., 13.02..., -0.02..., -0.70..., -1.37...]
sage: [round(A[:i,:i].determinant()) for i in range(1, A.nrows()+1)]
372 Chapter 19. Dense matrices using a NumPy backend.