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