Semidefinitii

Semidefinitii refers to the mathematical concept of semidefiniteness, most commonly in the context of symmetric (or Hermitian) matrices and quadratic forms. A real symmetric matrix A is called positive semidefinite if x^T A x ≥ 0 for all real vectors x. If A is negative semidefinite, then x^T A x ≤ 0 for all x. In the complex case, the analogous condition uses x^* A x with A Hermitian.

Equivalent characterizations include: all eigenvalues of A are nonnegative for positive semidefinite matrices; there exists a matrix B with A = B^T B (real case) or A = B^* B (complex case); and A has a unique positive semidefinite square root A^(1/2) such that A = A^(1/2) A^(1/2). For a semidefinite matrix, x^T A x = 0 if and only if x lies in the nullspace of A. Positive definite matrices are the stricter case where x^T A x > 0 for all nonzero x, implying all eigenvalues are strictly positive.

The set of positive semidefinite matrices forms a convex cone, closed under addition and nonnegative scalar multiplication. Sylvester’s criterion gives a practical test for positive definiteness of real symmetric matrices via leading principal minors.

Applications of semidefinite concepts are widespread. Semidefinite programming (SDP) generalizes linear programming to optimization over the cone of PSD matrices and is used in control theory, structural design, statistics (notably covariance matrices), signal processing, and machine learning (kernel methods). Semidefinite matrices naturally arise as covariance matrices, Gram matrices, and in stability analyses, underscoring their central role in both theory and applications.