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: 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 A Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.
be the leading principal minor matrices, i.e. the
upper left corner matrices.
is positive definite, then the principal minors are positive; that is,
since the determinant is the product of the eigenvalues.
To prove the reverse implication, we use induction.
Suppose the criterion holds for
Assuming that all the principal minors of
is positive definite by the inductive hypothesis.
Denote then By completing the squares, this last expression is equal to where
The first term is positive by the inductive hypothesis.
We now examine the sign of the second term.
By using the block matrix determinant formula on
Essentially the same proof as for the case that
is strictly positive definite shows that all principal minors (not necessarily the leading principal minors) are non-negative.
For the reverse implication, it suffices to show that if
has all non-negative principal minors, then for all t>0, all leading principal minors of the Hermitian matrix
is the nxn identity matrix.
Indeed, from the positive definite case, we would know that the matrices
are strictly positive definite.
Since the limit of positive definite matrices is always positive semidefinite, we can take
be the kth leading principal submatrix of
We use the identity in Characteristic polynomial#Properties to write
is the trace of the jth exterior power of
From Minor_(linear_algebra)#Multilinear_algebra_approach, we know that the entries in the matrix expansion of
In particular, the diagonal entries are the principal minors of
Since the trace of a matrix is the sum of the diagonal entries, it follows that