Testing for Positive Definiteness

Linear Algebra for Machine Learning · Axiom Academy

Testing for Positive Definiteness Three methods with worked examples A symmetric matrix is positive definite if and only if all eigenvalues are positive. Method 2: Sylvester's Criterion Check that all leading principal minors (determinants of upper-left sub-matrices) are positive. Directly test x^T A x > 0 for various vectors x. If this holds for enough vectors, A is positive definite. Eigenvalue test: Most reliable, gives you the actual eigenvalues. Useful for analysis but can be computationally expensive for large matrices. Sylvester's criterion: Fast for small matrices, purely deterministic. Good for hand calculations. Quadratic form test: Quick sanity check. Doesn't prove positive definiteness but can quickly show a matrix is NOT positive definite.

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