Covariance Matrix - Which Matrices Are Covariance Matrices?

Which Matrices Are Covariance Matrices?

From the identity just above, let be a real-valued vector, then

which must always be nonnegative since it is the variance of a real-valued random variable. and the symmetry of the covariance matrix's definition it follows that only a positive-semidefinite matrix can be a covariance matrix. The answer to the converse question, whether every symmetric positive semi-definite matrix is a covariance matrix, is "yes." To see this, suppose M is a p×p positive-semidefinite matrix. From the finite-dimensional case of the spectral theorem, it follows that M has a nonnegative symmetric square root, that can be denoted by M1/2. Let be any p×1 column vector-valued random variable whose covariance matrix is the p×p identity matrix. Then