1.5 SVD and low-rank approximation
You can compress a matrix and know what you threw away.
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The singular value decomposition factorises any matrix — square or not — into rotate, stretch, rotate, and truncating it gives the best possible low-rank approximation, with a guarantee about what was lost. It is the workhorse behind PCA, recommender factorisation and LoRA fine-tuning. It sits at the summit of the linear algebra sequence because it composes everything before it. The common confusion is thinking SVD needs a square matrix; its whole value is that it does not.
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The SVD as rotate–stretch–rotate
Every matrix, of any shape, can be read as a rotation, then a stretch along axes, then another rotation. This is the most generally useful decomposition in applied linear algebra.
Truncated SVD and the Eckart–Young theorem
Keeping only the largest stretch factors gives the best possible approximation of a matrix at that size, and there is a theorem saying so. Compression, denoising and recommendation all rest on this result.
Applications: PCA, latent semantic analysis, LoRA
The same decomposition underlies dimensionality reduction, one classical text-analysis method, and the low-rank adapters used to fine-tune large models. Seeing one technique behind all three is the point of this item.
Condition number and numerical stability
The ratio between the largest and smallest stretch factors predicts how much a small change in input can distort the output. It is the number that tells you whether to trust a numerical result.
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