svd(): singular value decomposition

svd(A) decomposes a matrix into orthogonal factors and singular values. It exposes rank, dominant directions, and numerical conditioning.

What it provides

Conceptually, SVD writes A as U × S × Vᵀ. The diagonal singular values show how strongly the matrix acts along independent directions. Small values can signal near-dependence in data.

Applications

Use SVD for dimensionality reduction, least-squares fitting, and low-rank approximation. Related tools include pinv(), transpose(), multiply(), qr(), and det().

Caveat

SVD is computationally expensive for large matrices. Choose a tolerance deliberately when deciding which singular values count as zero.