Overview

This is the deepest topic in the unit — the direct payoff of Topic 4's spectral theorem. Once every symmetric matrix is known to diagonalise with real eigenvalues (unconditionally), a natural refinement follows immediately: what if those eigenvalues are further constrained to be positive? This single sign condition turns out to be equivalent to five or six superficially different-looking properties — a quadratic form that's always positive, a matrix that factors as $A=LL^\top$, a matrix that is literally $S^\top S$ for some $S$ — and this topic proves the equivalences, builds the Cholesky factorisation that makes positive-definiteness computationally checkable, and closes with the Oxford M1 LA II capstone: classifying conics and quadric surfaces by orthogonal diagonalisation.