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:
In other words, all of the leading principal minors must be positive.
An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the leading principal minors:
a Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.

Proof

The proof is only for nonsingular Hermitian matrix with coefficients in, therefore only for nonsingular real-symmetric matrices.
Positive definite or semidefinite matrix: A symmetric matrix A whose eigenvalues are positive is called positive definite, and when the eigenvalues are just nonnegative, A is said to be positive semidefinite.
Theorem I: A real-symmetric matrix A has nonnegative eigenvalues if and only if A can be factored as A = BTB, and all eigenvalues are positive if and only if B is nonsingular.
Theorem II : The symmetric matrix A possesses positive pivots if and only if A can be uniquely factored as A = RTR, where R is an upper-triangular matrix with positive diagonal entries. This is known as the Cholesky decomposition of A, and R is called the Cholesky factor of A.
Theorem III: Let Ak be the k × k leading principal submatrix of An×n. If A has an LU factorization A = LU, where L is a lower triangular matrix with a unit diagonal, then det = u11u22 · · · ukk, and the k-th pivot is ukk = det = a11 for k = 1, ukk = det/det for k = 2, 3,..., n, where ukk is the -th entry of U for all k = 1, 2,..., n.
Combining Theorem II with Theorem III yields:
Statement I: If the symmetric matrix A can be factored as A=RTR where R is an upper-triangular matrix with positive diagonal entries, then all the pivots of A are positive, therefore all the leading principal minors of A are positive.
Statement II: If the nonsingular n × n symmetric matrix A can be factored as, then the QR decomposition of B yields:, where Q is orthogonal matrix and R is upper triangular matrix.
As A is non-singular and, it follows that all the diagonal entries of R are non-zero. Let rjj be the -th entry of E for all j = 1, 2,..., n. Then rjj ≠ 0 for all j = 1, 2,..., n.
Let F be a diagonal matrix, and let fjj be the -th entry of F for all j = 1, 2,..., n. For all j = 1, 2,..., n, we set fjj = 1 if rjj > 0, and we set fjj = -1 if rjj < 0. Then, the n × n identity matrix.
Let S=FR. Then S is an upper-triangular matrix with all diagonal entries being positive. Hence we have, for some upper-triangular matrix S with all diagonal entries being positive.
Namely Statement II requires the non-singularity of the symmetric matrix A.
Combining Theorem I with Statement I and Statement II yields:
Statement III: If the real-symmetric matrix A is positive definite then A possess factorization of the form A = BTB, where B is nonsingular, the expression A = BTB implies that A possess factorization of the form A = RTR where R is an upper-triangular matrix with positive diagonal entries, therefore all the leading principal minors of A are positive.
In other words, Statement III proves the "only if" part of Sylvester's Criterion for non-singular real-symmetric matrices.
Sylvester's Criterion: The real-symmetric matrix A is positive definite if and only if all the leading principal minors of A are positive.