Sylvester's criterion

Sylvester's criterion, a characterization of positive-definite Hermitian matrices.

This condition is known as Sylvester's criterion, and provides an efficient test of positive-definiteness of a symmetric real matrix.

Applying this argument to the minors of , the positive definiteness follows by Sylvester's criterion.)

In mathematics, Sylvester's criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite.

Sylvester's criterion states that a Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant:

Sylvester's Criterion: The real-symmetric matrix A is positive definite if and only if all the leading principal minors of A are positive.

Since the kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row k, Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive.