Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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