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.
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