principal minor

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

All the leading principal minors of A are positive.

In the two variable case, and are the principal minors of the Hessian.

This is a positive self-adjoint operator so its principal minors do not vanish.

Its leading principal minors are all positive.

On matrices with non-positive off-diagonal elements and positive principal minors.

Compute each principal minor of that matrix.

For an n x n square matrix, there are n leading principal minors.

It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries 0 and -1.

This trace may be computed as the sum of all principal minors of A of size k.

A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative.

(the sum of principal minors)

The condition of positivity of all principal minors is also known as the Hawkins-Simon condition in economic literature.

Applying this argument to the principal minors of , the positive semidefiniteness follows by Sylvester's criterion.)

Furthermore, if the principal minors of the matrix are all positive (known as the Hawkins-Simon condition), the required output vector is non-negative.

A sufficient matrix is a generalization both of a positive-definite matrix and of a P-matrix, whose principal minors are each positive.

The kth leading principal minor of a matrix M is the determinant of its upper-left k by k sub-matrix.

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 matrix is negative definite if its k-th order leading principal minor is negative when k is odd, and positive when k is even.

If is invertible, then it admits an LU (or LDU) factorization if and only if all its leading principal minors are nonsingular.

The ith Hurwitz determinant is the determinant of the ith leading principal minor of the above Hurwitz matrix H.

If is a singular matrix of rank , then it admits an LU factorization if the first leading principal minors are nonsingular, although the converse is not true.

Since the kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row k, Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive.

For Hermitian matrices, the leading principal minors can be used to test for positive definiteness and the principal minors can be used to test for positive semidefiniteness.

Specifically, the sufficient condition for a minimum is that all of these principal minors be positive, while the sufficient condition for a maximum is that the minors alternate in sign with the 1x1 minor being negative.