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TL;DR

Any matrix A can be decomposed as \( A = U\Sigma V^T \) where U and V are orthogonal and \( \Sigma \) is diagonal with non-negative singular values. It generalizes eigendecomposition to rectangular matrices.

By Valenke Exam Prep Team·Last updated 2026-06-03

Singular Value Decomposition

Any matrix A can be decomposed as \( A = U\Sigma V^T \) where U and V are orthogonal and \( \Sigma \) is diagonal with non-negative singular values. It generalizes eigendecomposition to rectangular matrices.

Why it matters for interviews

SVD is used for low-rank approximation (noise filtering), pseudoinverse computation (regression), and latent factor models. Understanding SVD demonstrates strong linear algebra skills valued in quant interviews.

Definition and Mathematical Foundation

Any matrix A can be decomposed as \( A = U\Sigma V^T \) where U and V are orthogonal and \( \Sigma \) is diagonal with non-negative singular values. It generalizes eigendecomposition to rectangular matrices.

Application in Quantitative Finance

SVD is used for low-rank approximation (noise filtering), pseudoinverse computation (regression), and latent factor models. Understanding SVD demonstrates strong linear algebra skills valued in quant interviews.

Related Terms

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Frequently Asked Questions

How does SVD relate to PCA?▼
PCA of centered data X can be computed via SVD of X directly: \( X = U\Sigma V^T \). The right singular vectors V are the principal components, and the singular values squared (divided by n-1) give eigenvalues of the covariance matrix.
What is the Eckart-Young theorem?▼
The best rank-k approximation to A (in Frobenius or spectral norm) is obtained by keeping only the top k singular values and vectors. This is used for noise reduction and data compression.
How is SVD used in recommender systems and factor models?▼
SVD decomposes a user-item (or asset-factor) matrix into latent factors. In finance, this identifies hidden market factors driving asset returns, similar to statistical arbitrage factor models.