Positive-definite matrix

In linear algebra, a symmetric n × n real matrix M is said to be positive definite if z^TMz is positive for every non-zero columnvector
z of n real numbers. Here z^T denotes thetranspose of
z.

• The real symmetric matrix

is positive definite since for any non-zero column vector z with entriesa,
b and c, we have

This result is a sum of squares, and therefore non-negative; and is zero only ifa =
b = c = 0, that is, when z is zero.

• The real symmetric matrix

is not positive definite. If z is the vector , one has

More generally, an n × n Hermitian matrix M is said to be positive definite if
z*Mz is real and positive for all non-zero complex vectors z. Here
z* denotes the conjugate transpose of z.

摘自：https://en.wikipedia.org/wiki/Positive_semidefinite_matrix