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 n × n 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,
• ${\displaystyle {}\quad \vdots }$
• M itself.

In other words, all of the leading principal minors must be positive. By using appropriate permutations of rows and columns of M, it can also be shown that the positivity of any nested sequence of n principal minors of M is equivalent to M being positive-definite.^[1]

An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the leading principal minors as illustrated by the Hermitian matrix

${\displaystyle {\begin{pmatrix}0&0&-1\\0&-1&0\\-1&0&0\end{pmatrix}}\quad {\text{with eigenvectors}}\quad {\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad {\begin{pmatrix}1\\0\\1\end{pmatrix}}\quad {\text{and}}\quad {\begin{pmatrix}1\\0\\-1\end{pmatrix}}.}$

A Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.^[2][3]