Wat is LU, Cholesky, QR and singular value decompositions?

The LU factorisation writes $PA = LU$ with a permutation $P$ (partial pivoting), unit lower-triangular $L$ and upper-triangular $U$. The Cholesky factorisation of a symmetric positive-definite matrix is $A = LL^\top$ with $L$ lower triangular. The QR factorisation writes $A = QR$ with orthonormal columns in $Q$ and $R$ upper triangular. The singular value decomposition writes $A = U\Sigma V^\top$ with orthonormal $U$, $V$ and a nonnegative diagonal $\Sigma$. The tridiagonal matrix algorithm (Thomas’s algorithm) solves a tridiagonal system by one forward elimination and one back substitution, in $O(n)$ operations.