Semidefiniittisyys

Semidefiniittisyys is a property of matrices and quadratic forms. A real symmetric matrix A is said to be positive semidefinite if for every non-zero real vector x, the quadratic form x^T Ax is non-negative. This means that x^T Ax >= 0 for all x. A matrix that is positive semidefinite is also said to be negative semidefinite if the quadratic form x^T Ax is non-positive for all non-zero real vectors x. This means that x^T Ax <= 0 for all x.

A matrix that is neither positive semidefinite nor negative semidefinite is indefinite. The concept of semidefiniteness is crucial in various fields, including optimization, linear algebra, and statistics. For example, in optimization, the Hessian matrix of a function must be positive semidefinite at a point for that point to be a local minimum.

There are several ways to check for semidefiniteness. One common method is to examine the eigenvalues of the matrix. A real symmetric matrix is positive semidefinite if and only if all of its eigenvalues are non-negative. Similarly, it is negative semidefinite if and only if all of its eigenvalues are non-positive. Another method involves checking the principal minors of the matrix. For a positive semidefinite matrix, all leading principal minors must be non-negative.