Every symmetric matrix diagonalises with real eigenvalues, unconditionally, by Corollary 1 of the spectral-theorem note. This note studies the refinement that matters most in practice: what if those real eigenvalues are further guaranteed non-negative, or strictly positive? This single sign condition — positive semi-definiteness or positive definiteness — turns out to have (at least) five equivalent faces, proved here, each the natural test to reach for in a different situation.
Learning objectives. By the end of this note you should be able to:
- state the definition of a positive (semi-)definite operator/matrix;
- prove the equivalence between the eigenvalue test and the quadratic-form test;
- prove that is always positive semi-definite, for any ;
- recognise why a covariance matrix is always positive semi-definite.
In this lesson
- Definition and the equivalence theorem
Before you start
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