semidefiniten

Semidefiniten is a mathematical term referring to a property of square matrices. A real symmetric or complex Hermitian matrix is called positive semidefinite if all its eigenvalues are non-negative. Similarly, a matrix is negative semidefinite if all its eigenvalues are non-positive.

The concept is crucial in various fields of mathematics, including linear algebra, optimization, and functional analysis. For instance, in optimization, the Hessian matrix of a function must be positive semidefinite for the function to have a local minimum at a critical point.

A matrix is considered semidefinite if it is either positive semidefinite or negative semidefinite. This means that for any non-zero vector x, the quadratic form xᵀAx (for real symmetric A) or xAx (for complex Hermitian A) has the same sign (or is zero) for all x. If the quadratic form is always non-negative, the matrix is positive semidefinite. If it is always non-positive, the matrix is negative semidefinite.

If a matrix is semidefinite, it possesses certain structural properties. For example, the null space of a positive semidefinite matrix is the same as the null space of its corresponding positive definite part. Semidefinite matrices play a significant role in semidefinite programming, a subfield of convex optimization.