Four matrix factorizations — LU, QR, SVD, Cholesky — explained in one chart
burny_tech · x · 2026-11-11
A clean visual explainer of the four factorizations that cover most of numerical linear algebra:
- LU with pivoting (PA=LU): Gaussian elimination recorded as matrices; solves square systems Ax=b with P reordering rows for stability.
- QR: splits A into an orthogonal Q and upper triangular R; the numerically stable route to least squares.
- SVD (A=UΣVᵀ): exists for every real matrix, exposes rank and principal directions; truncating the largest singular values yields the best low-rank approximation, underpinning PCA and compression.
- Cholesky (A=LLᵀ): only for symmetric positive definite matrices, at roughly half the cost of LU.
The core idea: rewrite one hard matrix into structured pieces that are easier to solve with, fit, or compress.
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