Sylvester's Criterion

Sylvester's Criterion

In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester.

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 ,
• the upper left 2-by-2 corner of ,
• the upper left 3-by-3 corner of ,
• ...
• itself.

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