Why every diagonal entry of positive definite matrix is positive? Solution A matrix M is positive-semidefinite if and only if it arises as the Gram matrix of some set of vectors. In contrast to the positive-definite case, these vectors need not be linearly independent. For any matrix A, the matrix A*A is positive semidefinite, and rank(A) = rank(A*A). Conversely, any Hermitian positive semi-definite matrix M can be written as M = LL*, where L is lower triangular; this is the Cholesky decomposition. If M is not positive definite, then some of the diagonal elements of L may be zero. A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries 0 and -1..