Positive-definite Matrix

Positive-definite Matrix

In linear algebra, a symmetric n × n real matrix M is said to be positive definite if zTMz is positive, for all non-zero column vectors z of n real numbers; where zT denotes the transpose of z.

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

The negative definite, positive semi-definite, and negative semi-definite matrices are defined in the same way, except that the formula zTMz or z*Mz is required to be always negative, non-negative, and non-positive, respectively.

Positive definite matrices are closely related to a positive-definite symmetric bilinear forms (or sesquilinear forms in the complex case), and to inner products of vector spaces.

Some authors use more general definitions of "positive definite" that include some non-symmetric real matrices, or non-Hermitian complex ones. However, some of those extended definitions are incompatible, and may not be suitable for certain applications.