- - - figure the eigenvectors and eigenvalues of aadetermine the principal axes of the ellipse the following definition generalizes the notion of positivity of real variable to that of (hermitianmatrixproviding crucial concept for multivariate differentiation and optimizationsee appendix definition positive (semi)definite matrix hermitian matrix is called positive semidefinite (we write if haxxi for all it is called positive definite (we write if haxxi for all the positive (semi)definiteness of matrix can be directly related to the positivity of its eigenvaluesas followstheorem eigenvalues of positive semidefinite matrix all eigenvalues of positive semidefinite matrix are non-negative and all eigenvalues of positive definite matrix are strictly positive prooflet be positive semidefinite matrix by theorem the eigenvalues of are all real suppose is an eigenvalue with eigenvector as is positive semidefinitewe have havvi hvvi kvk which can only be true if similarlyfor positive definite matrixl must be strictly greater than corollary any real positive semidefinite matrix can be written as bbfor some real matrix converselyfor any real matrix bthe matrix bbis positive semidefinite positive semidefinite