to be negative-definite if for all non-zero in . Formally, An symmetric real matrix is said to be negative-semidefinite or non-positive-definite if for all in . Formally, An symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called indefinite. Definitions for complex matrices The following definitions all involve the term . Notice that this is always a real number for any Hermitian square matrix . An Hermitian complex matrix is said to be positive-definite if for all non-zero in . Formally, An Hermitian complex matrix is said to be positive semi-definite or non-negative-definite if for all in . Formally, An Hermitian complex matrix is said to be negative-definite if for all non-zero in . Formally, An Hermitian complex matrix is said to be negative semi-definite or non-positive-definite if for all in . Formally, An Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called indefinite. Consistency between real and complex definitions Since every real matrix is also a complex matrix, the definitions of "definiteness" for the two classes must agree. For complex matrices, the most common definition says that is positive-definite if and only if is real and positive for every non-zero complex column vectors . This condition implies that is Hermitian (i.e. its transpose is equal to its conjugate), since being real, it equals its conjugate transpose for every which implies . By this definition, a positive-definite real matrix is Hermitian, hence symmetric; and is positive for all non-zero real column vectors . However the last condition alone is not sufficient for to be positive-definite. For example, if then for any real vector with entries and we have , which is always positive if is not zero. However, if is the complex vector with entries and , one gets which is not real. Therefore, is not positive-definite. On the other hand, for a symmetric real matrix , the condition " for all nonzero real vectors " does imply that is positive-definite in the complex sense. Notation If a Hermitian matrix is positive semi-definite, one sometimes writes and if is positive-definite one writes . To denote that is negative semi-definite one writes and to denote that is negative-definite one writes . The notion comes from functional analysis where positive semidefinite matrices define positive operators. If two matrices and satisfy , we can define a non-strict partial order that is reflexive, antisymmetric, and transitive; It