The map of factorizations

Eight factorizations, and the one question each answers:

Method	Works on	What it gives you
LU	Square matrix	Gaussian elimination, saved for reuse
QR	Any matrix	Perpendicular, unit-length directions
Eigendecomposition	Square matrix	Directions that only get scaled
SVD	Any matrix	The most general matrix factorization
PCA	Data matrix	Compression to fewer features
Pseudoinverse	Any matrix	An “inverse” when no true inverse exists
Cholesky	Symmetric positive-definite	A “square root” of a covariance matrix
Tucker / CP	Tensor, any order	PCA generalized to every axis

Everything above the last row works on matrices — two axes. Real data often has more. That is what section 10 addresses.