Semidefiniert

Semidefiniert is a term used in mathematics, particularly in linear algebra and optimization, to describe a property of matrices and quadratic forms. A real symmetric matrix $A$ is said to be semidefiniert, or more specifically positive semidefinite or negative semidefinite, if it satisfies certain conditions related to its eigenvalues or the values of the quadratic form it defines.

A symmetric matrix $A$ is positive semidefinite if for every non-zero vector $x$, the quadratic form $x^T Ax$ is greater than or equal to zero. This is equivalent to saying that all eigenvalues of $A$ are non-negative. Similarly, a symmetric matrix $A$ is negative semidefinite if for every non-zero vector $x$, the quadratic form $x^T Ax$ is less than or equal to zero. This is equivalent to saying that all eigenvalues of $A$ are non-positive.

Matrices that are neither positive semidefinite nor negative semidefinite are called indefinite. If a matrix is both positive semidefinite and negative semidefinite, it must be the zero matrix, in which case all eigenvalues are zero. The term "semidefiniert" often implicitly refers to either positive semidefinite or negative semidefinite, depending on the context. These concepts are fundamental in areas such as convex optimization, where the positive semidefiniteness of the Hessian matrix is a condition for convexity.