Six matrix factorizations dominate in numerical linear algebra and matrix analysis: for most purposes one of them is sufficient for the task at hand. - The Big Six Matrix Factorizations / HN

Cholesky Factorization - solving positive definite linear systems.

LU Factorization - solving general linear systems.

QR Factorization - solving least squares problems, computing an orthonormal basis for the range space of A, orthogonalization.

Schur Decomposition - computing eigenvalues and eigenvectors, computing invariant subspaces, evaluating matrix functions.

Spectral Decomposition - any problem involving eigenvalues of Hermitian matrices.

Singular Value Decomposition - determining matrix rank, solving rank-deficient least squares problems, computing all kinds of subspace information.