Sylvester's criterion

Sylvester's criterion is a fast determinant test for positive definiteness: a real symmetric matrix B is positive definite if and only if all of its leading principal minors are strictly positive. The leading principal minor of order k is the determinant of the top-left k-by-k corner of B, so you check det of the 1x1 corner, then the 2x2 corner, up to the full n-by-n determinant — and they must ALL be positive.

The appeal is that you never compute an eigenvalue. Eigenvalues require root-finding; leading minors require only n determinants, and there is a slick recursion (the determinants relate to the pivots in symmetric Gaussian elimination, all of which must be positive). It is the workhorse check when you must certify positive definiteness by hand or quickly in code.

Two warnings keep people honest. First, the test is for positive definiteness only; for negative definiteness you apply it to -B, which makes the leading minors of B alternate in sign starting negative. Second — the famous trap — checking minors does NOT extend to the semidefinite case. A matrix can have all leading principal minors >= 0 and still fail to be positive semidefinite; the standard counterexample is [0, 0; 0, -1], whose leading minors are 0 and 0 yet which is not semidefinite.

The correct semidefinite version uses ALL principal minors, not just the leading (top-left nested) ones: B is positive semidefinite iff every principal minor — the determinant of any symmetric selection of rows and columns — is nonnegative. That is far more work, which is exactly why the clean leading-minor shortcut is reserved for the strict positive-definite case.