Is there a criterion for positive semidefiniteness of a matrix in terms of dimension reduction, i.e, such that positive semi-definiteness of $n \times n$ matrix is expressed as positive semidefiniteness of smaller matrices and possibly some additional condition?
Could anyone give me hint? Thanks for replies.
UPD: besides the version of Sylvester's criterion for semi-definite case.
Best Answer
An alternative approach to modified Sylvester's criterion has been given under a related question. It is a recursive approach based on row reduction (or Gaussian elimination).
(Restated below for completeness.)
For a $n\times n$ Hermitian (symmetric) matrix $A$: