Mathematics · Core
Singular Value Decomposition
SVD factors any matrix A = U Σ V' into orthogonal rotations and a diagonal of singular values — the workhorse behind PCA, low-rank approximation, and many recommenders.
Definition
U and V are orthogonal; Σ is nonnegative diagonal. The best rank-k approximation (Eckart–Young) keeps the top k singular values. PCA on a data matrix is an SVD of the (centered) data.
Why it matters
It is how you actually compute PCA and how you compress a panel. Condition numbers of Σ tell you whether inversion is sane.
Case
A 400-name residual-covariance estimated from 80 days is rank-deficient; SVD plus shrinkage is the difference between a number and a NaN in a risk inverse.
How to read it
Look at the singular-value scree. A flat tail means you should not invert the raw matrix.
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