Definite matrix information

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

Positive semi-definite matrices are defined similarly, except that the scalars $z^{\operatorname {T} }Mz$ and $z^{*}Mz$ 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 $B$ with conjugate transpose $B^{*}$ such that $M=B^{*}B.$

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.

^ "Appendix C: Positive Semidefinite and Positive Definite Matrices". Parameter Estimation for Scientists and Engineers: 259–263. doi:10.1002/9780470173862.app3.
^ Boyd, Stephen; Vandenberghe, Lieven (2004-03-08). Convex Optimization. Cambridge University Press. ISBN 978-0-521-83378-3.

[1] "Appendix C: Positive Semidefinite and Positive Definite Matrices". Parameter Estimation for Scientists and Engineers: 259–263. doi:10.1002/9780470173862.app3.

[2] Boyd, Stephen; Vandenberghe, Lieven (2004-03-08). Convex Optimization. Cambridge University Press. ISBN 978-0-521-83378-3.

Definite matrix information

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Definite matrix

Square matrix

Square root of a matrix

Positive definiteness

Hessian matrix

Definite quadratic form

Cholesky decomposition

Matrix decomposition

Polar decomposition

Square root of a 2 by 2 matrix

Hermitian matrix

Eigendecomposition of a matrix

Stiffness matrix

Symmetric matrix

Covariance matrix

Invertible matrix

Negative definiteness

Wishart distribution

Schur product theorem

Shear mapping

Determinant

Estimation of covariance matrices

Scatter matrix

Jacobi method