semidefinitie

Semidefinitie is a term originating from linear algebra and matrix theory, describing a specific class of symmetric matrices based on their eigenvalues and quadratic forms. A real symmetric matrix \(A\) is called semidefinite if it meets certain positivity or negativity conditions.

A matrix \(A\) is said to be positive semidefinite (often abbreviated as PSD) if, for all vectors \(x\), the quadratic form \(x^T A x \geq 0\). Equivalently, all eigenvalues of \(A\) are non-negative. Positive semidefinite matrices are fundamental in various fields such as optimization, statistics, and engineering, especially in contexts involving covariance matrices, kernel methods, and quadratic programming.

Conversely, a matrix is negative semidefinite if \(x^T A x \leq 0\) for all \(x\), with all eigenvalues being non-positive. Semidefinite matrices are closely related to definiteness: if \(A\) is positive or negative semidefinite, it is called semidefinite; if it is strictly positive or negative definite (i.e., all eigenvalues are strictly positive or negative), it is simply termed positive definite or negative definite, respectively.

Semidefinite matrices are characterized by the existence of a Cholesky or spectral decomposition. They are also significant in optimization, particularly in semidefinite programming, where problems are formulated as optimizing a linear function subject to the constraint that a particular matrix variable remains semidefinite.

Understanding the properties of semidefinite matrices is essential in numerous theoretical and applied disciplines, providing a foundation for stability analysis, statistical modeling, and convex optimization.