| # Semidefinite cone ## Positive semidefinite matrices A symmetric matrix is said to be positive semi-definite if the associated quadratic form is non-negative, that is: An alternate condition is that every eigenvalue of is non-negative. We use the acronym PSD to refer to the term ‘‘positive semidefinite’’, and the notation to express that is PSD. A matrix is said to be positive definite if the above condition is satisfied strictly for non-zero : An alternate condition is that every eigenvalue of is positive. We use the acronym PD to refer to the term ‘‘positive definite’’, and the notation to express that is PD. ## Semidefinite cone The set of PSD matrices in is denoted . That of PD matrices, . The set is a convex cone, called the semidefinite cone. The fact that it is convex derives from its expression as the intersection of half-spaces in the subspace of symmetric matrices. Indeed, we have ## Rank-one PSD matrices PSD matrices with rank one can be expressed as for some . The associated quadratic form is a squared linear form: ## Link with covariance matrices Covariance matrices are PSD matrices, since they can be expressed as an expected value of a squared linear form: if is a random variable, the covariance matrix of is defined as where denotes the expectation operator. Conversely, any PSD matrix can be interpreted as a covariance matrix, for some distribution. Hence, the PSD cone is exactly the set of covariance matrices.