Semidefinite cone

Positive semidefinite matrices

A symmetric matrix P = P^T in mathbf{R}^{n times n} is said to be positive semi-definite if the associated quadratic form is non-negative, that is:

 forall : x in mathbf{R}^n ~:~ x^TPx ge 0.

An alternate condition is that every eigenvalue of P is non-negative. We use the acronym PSD to refer to the term ‘‘positive semidefinite’’, and the notation P succeq 0 to express that P is PSD.

A matrix is said to be positive definite if the above condition is satisfied strictly for non-zero x:

 forall : x in mathbf{R}^n, ;; x ne 0 ~:~ x^TPx > 0.

An alternate condition is that every eigenvalue of P is positive. We use the acronym PD to refer to the term ‘‘positive definite’’, and the notation P succ 0 to express that P is PD.

Semidefinite cone

The set of PSD matrices in mathbf{R}^{n times n} is denoted mathbf{S}_+. That of PD matrices, mathbf{S}_{++}.

The set mathbf{S}_+ 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 mathbf{S}^n of symmetric matrices. Indeed, we have

 mathbf{S}_+ = bigcap_{x in mathbf{R}^n} left{ P in mathbf{S}^n ~:~ x^TPx ge 0 right} .

Rank-one PSD matrices

PSD matrices with rank one can be expressed as P = vv^T for some v in mathbf{R}^n. The associated quadratic form is a squared linear form:

 x^TPx = x^T(vv^T)x = (x^Tv) (v^Tx) = (v^Tx)^2.

Link with covariance matrices

Covariance matrices are PSD matrices, since they can be expressed as an expected value of a squared linear form: if X is a random variable, the covariance matrix of X is defined as

 mathbf{E} (X - mathbf{E}(X))(X - mathbf{E}(X))^T,

where mathbf{E} 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.