Definite matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a symmetric matrix 
 
 
 
 M
 
 
 {\displaystyle M}
 
 with <a href="/facts/Real_number/R02gw5Pb">real</a> entries is positive-definite if the real number 
 
 
 
 
 
 x
 
 
 ⊤
 
 
 M
 
 x
 
 
 
 {\displaystyle \mathbf {x} ^{\top }M\mathbf {x} }
 
 is positive for every nonzero real <a href="/facts/Column_vector/pPuRr9Xk">column vector</a> 
 
 
 
 
 x
 
 ,
 
 
 {\displaystyle \mathbf {x} ,}
 
 where 
 
 
 
 
 
 x
 
 
 ⊤
 
 
 
 
 {\displaystyle \mathbf {x} ^{\top }}
 
 is the <a href="/facts/Row_vector/pPuRr9Xk">row vector</a> <a href="/facts/Transpose/8wmsagGS">transpose</a> of 
 
 
 
 
 x
 
 .
 
 
 {\displaystyle \mathbf {x} .}

More generally, a <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a> (that is, a <a href="/facts/Complex_matrix/qa8FI4ko">complex matrix</a> equal to its <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>) is positive-definite if the real number 
 
 
 
 
 
 z
 
 
 ∗
 
 
 M
 
 z
 
 
 
 {\displaystyle \mathbf {z} ^{*}M\mathbf {z} }
 
 is positive for every nonzero complex column vector 
 
 
 
 
 z
 
 ,
 
 
 {\displaystyle \mathbf {z} ,}
 
 where 
 
 
 
 
 
 z
 
 
 ∗
 
 
 
 
 {\displaystyle \mathbf {z} ^{*}}
 
 denotes the conjugate transpose of 
 
 
 
 
 z
 
 .
 
 
 {\displaystyle \mathbf {z} .}

Positive semi-definite matrices are defined similarly, except that the scalars 
 
 
 
 
 
 x
 
 
 ⊤
 
 
 M
 
 x
 
 
 
 {\displaystyle \mathbf {x} ^{\top }M\mathbf {x} }
 
 and 
 
 
 
 
 
 z
 
 
 ∗
 
 
 M
 
 z
 
 
 
 {\displaystyle \mathbf {z} ^{*}M\mathbf {z} }
 
 are required to be positive or zero (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite.
Some authors use more general definitions of definiteness, permitting the matrices to be non-symmetric or non-Hermitian. The properties of these generalized definite matrices are explored in § Extension for non-Hermitian square matrices, below, but are not the main focus of this article.

Definite matrix open-in-new

Definite matrix