In mathematics, Sylvester's criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite, due to James Joseph Sylvester. 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. 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
A Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.
Proof for the case of positive definite matrices Suppose M n {\displaystyle M_{n}} is n × n {\displaystyle n\times n} Hermitian matrix M n † = M n {\displaystyle M_{n}^{\dagger }=M_{n}} . Let M k , k = 1 , … n {\displaystyle M_{k},k=1,\ldots n} be the leading principal minor matrices, i.e. the k × k {\displaystyle k\times k} upper left corner matrices. It will be shown that if M n {\displaystyle M_{n}} is positive definite, then the principal minors are positive; that is, det M k > 0 {\displaystyle \det M_{k}>0} for all k {\displaystyle k} .
M k {\displaystyle M_{k}} is positive definite. Indeed, choosing
we can notice that 0 < x † M n x = x → † M k x → . {\displaystyle 0<x^{\dagger }M_{n}x={\vec {x}}^{\dagger }M_{k}{\vec {x}}.} Equivalently, the eigenvalues of M k {\displaystyle M_{k}} are positive, and this implies that det M k > 0 {\displaystyle \det M_{k}>0} since the determinant is the product of the eigenvalues. To prove the reverse implication, we use induction. The general form of an ( n + 1 ) × ( n + 1 ) {\displaystyle (n+1)\times (n+1)} Hermitian matrix is
where M n {\displaystyle M_{n}} is an n × n {\displaystyle n\times n} Hermitian matrix, v → {\displaystyle {\vec {v}}} is a vector and d {\displaystyle d} is a real constant. Suppose the criterion holds for M n {\displaystyle M_{n}} . Assuming that all the principal minors of M n + 1 {\displaystyle M_{n+1}} are positive implies that det M n + 1 > 0 {\displaystyle \det M_{n+1}>0} , det M n > 0 {\displaystyle \det M_{n}>0} , and that M n {\displaystyle M_{n}} is positive definite by the inductive hypothesis. Denote
then
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