# Understanding Sylvester's Criterion determining positive definiteness

on Wikipedia the theorem states that

Sylvester's criterion states that a Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,

the upper left 2-by-2 corner of M,

the upper left 3-by-3 corner of M,

M itself.

In other words, all of the leading principal minors must be positive.

Now the definition of a principal minor is strange to me in this case, does this simply mean the entries along the diagonal must be positive for the matrix to be positive definite? is this definition the same for non negative elements along the diagonal meaning the matrix is positive semi-definite?

## 1 Answer

Yes, the diagonal entries of a positive definite matrix must be positive, and the diagonal entries of a positive semidefinite matrix must be nonnegative. But that's a necessary condition, not sufficient. You still need those determinants to be positive.

• How is it a necessary condition? because the determinant is the sum of the eigenvalues? Aug 20 '20 at 20:33
• The determinant is the product of the eigenvalues. But the reason the diagonal entries must be positive, if you think of positive definite as meaning $x^* M x > 0$ for every nonzero vector $x$, is that the diagonal entry $M_{ii}$ is $x^* M x$ where $x$ has a $1$ in position $i$ and $0$ everywhere else. Aug 20 '20 at 20:38