Definite matrix

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.

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