In mathematics, a symmetric matrix ${\displaystyle M}$ with real entries is positive-definite if the real number ${\displaystyle z^{\operatorname {T} }Mz}$ is positive for every nonzero real column vector ${\displaystyle z,}$ where ${\displaystyle z^{\operatorname {T} }}$ is the transpose of ${\displaystyle z}$.^[1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number ${\displaystyle z^{*}Mz}$ is positive for every nonzero complex column vector ${\displaystyle z,}$ where ${\displaystyle z^{*}}$ denotes the conjugate transpose of ${\displaystyle z.}$

Positive semi-definite matrices are defined similarly, except that the scalars ${\displaystyle z^{\operatorname {T} }Mz}$ and ${\displaystyle z^{*}Mz}$ 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.

A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines an inner product.

Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A matrix M is positive-definite if and only if it satisfies any of the following equivalent conditions.

• M is congruent with a diagonal matrix with positive real entries.
• M is symmetric or Hermitian, and all its eigenvalues are real and positive.
• M is symmetric or Hermitian, and all its leading principal minors are positive.
• There exists an invertible matrix ${\displaystyle B}$ with conjugate transpose ${\displaystyle B^{*}}$ such that ${\displaystyle M=B^{*}B.}$

A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed.

Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point p, then the function is convex near p, and, conversely, if the function is convex near p, then the Hessian matrix is positive-semidefinite at p.

The set of positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone.^[2]

Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.

Definitions

In the following definitions, ${\displaystyle \mathbf {x} ^{\operatorname {T} }}$ is the transpose of ${\displaystyle \mathbf {x} }$, ${\displaystyle \mathbf {x} ^{*}}$ is the conjugate transpose of ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {0} }$ denotes the n-dimensional zero-vector.

Definitions for real matrices

An ${\displaystyle n\times n}$ symmetric real matrix ${\displaystyle M}$ is said to be positive-definite if ${\displaystyle \mathbf {x} ^{\operatorname {T} }M\mathbf {x} >0}$ for all non-zero ${\displaystyle \mathbf {x} }$ in ${\displaystyle \mathbb {R} ^{n}}$. Formally,

An ${\displaystyle n\times n}$ symmetric real matrix ${\displaystyle M}$ is said to be positive-semidefinite or non-negative-definite if ${\displaystyle \mathbf {x} ^{\operatorname {T} }M\mathbf {x} \geq 0}$ for all ${\displaystyle \mathbf {x} }$ in ${\displaystyle \mathbb {R} ^{n}}$. Formally,

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