SPDMatrizen

SPDMatrizen, in English symmetric positive definite matrices, are real square matrices that are symmetric (A^T = A) and satisfy x^T A x > 0 for all nonzero vectors x. This condition is equivalent to A having strictly positive eigenvalues, or to the existence of a Cholesky factorization A = L L^T with a lower triangular L and positive diagonal entries. SPD matrices can also be written as A = Q Λ Q^T with Q orthogonal and Λ diagonal with positive entries, reflecting their positive-definite nature.

SPD matrices form a convex cone inside the space of symmetric matrices: if A and B are SPD and α, β ≥ 0 with α + β > 0, then αA + βB is SPD. They are always invertible, and many numerical methods exploit this structure. The Cholesky decomposition, A = L L^T, is widely used to solve linear systems efficiently. The square root A^{1/2} exists and is unique, and A^{−1} is also SPD. SPD matrices induce a weighted inner product and norm, defined by <x, y>_A = x^T A y and ||x||_A = sqrt(x^T A x), which in turn define metrics and distances such as the Mahalanobis distance.

Applications of SPD matrices are common in optimization, statistics, and numerical linear algebra. They define valid quadratic forms, ensuring convexity in optimization problems. Covariance matrices are SPD, and distance measures like Mahalanobis use the inverse of an SPD matrix. Classic examples include the identity matrix I_n, and the n×n tridiagonal matrix with 2 on the diagonal and −1 on the off-diagonals, both of which are SPD.