semidefinite symmetric matrices is convex. That is, if and are positive semidefinite, then for any between 0 and 1, is also positive semidefinite. For any vector : This property guarantees that semidefinite programming problems converge to a globally optimal solution. Relation with cosine The positive-definiteness of a matrix expresses that the angle between any vector and its image is always : Further properties If is a symmetric Toeplitz matrix, i.e. the entries are given as a function of their absolute index differences: , and the strict inequality holds, then is strictly positive definite. Let and Hermitian. If (resp., ) then (resp., ). If is real, then there is a such that , where is the identity matrix. If denotes the leading minor, is the kth pivot during LU decomposition. A matrix is negative definite if its k-th order leading principal minor is negative when is odd, and positive when is even. If is a real positive definite matrix, then there exists a positive real number such that for every vector , . 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. Block matrices and submatrices A positive matrix may also be defined by blocks: where each block is . By applying the positivity condition, it immediately follows that and are hermitian, and . We have that for all complex , and in particular for . Then A similar argument can be applied to , and thus we conclude that both and must be positive definite. The argument can be extended to show that any principal submatrix of is itself positive definite. Converse results can be proved with stronger conditions on the blocks, for instance, using the Schur complement. Local extrema A general quadratic form on real variables can always be written as where is the column vector with those variables, and is a symmetric real matrix. Therefore, the matrix being positive definite means that has a unique minimum (zero) when is zero, and is strictly positive for any other . More generally, a twice-differentiable real function on real variables has local minimum at arguments if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be