· 5 min
The matrices that act as squared lengths. Positive semidefinite matrices characterised by nonnegative eigenvalues, by a Gram factorisation, and by a square root, the Cholesky factorisation of a positive definite matrix, and the covariance matrix as the canonical example.
· 5 min
How a symmetric matrix is diagonalised. Eigenvalues and eigenvectors, the real spectrum and orthogonal eigenvectors of a symmetric matrix, the spectral theorem diagonalising it by an orthonormal basis, and the variational characterisation behind principal component analysis.
· 6 min
Geometry on a vector space. Inner products, the Cauchy-Schwarz inequality, the norm they induce, the parallelogram law that characterises inner-product norms, orthogonality with Pythagoras and Bessel's inequality, and the definition of a Hilbert space.