The cost table

Leading-term flop counts for a dense m × n factorization with m ≥ n, from Trefethen & Bau, Numerical Linear Algebra:

Method	Flops	The question it answers
Cholesky (n × n SPD)	n³/3	Solve Ax = b many times, A symmetric positive definite
LU (n × n)	2n³/3	The same, A merely square
QR (m × n)	2mn² − 2n³/3	Least squares, without forming XᵀX
Eigendecomposition (n × n sym.)	≈ 9n³	What does repeated application converge to?
Thin SVD (m × n)	2mn² + 11n³	The best rank-k approximation of anything
Randomized SVD (rank k)	≈ 4mnk	The same, when k ≪ n and A is dense
Lanczos (rank k, sparse)	O(k · nnz(A))	The same, when A is sparse

Cholesky is the cheapest and the fussiest: it needs symmetric positive definiteness, and says so.