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TL;DR

Factoring a matrix into a product of simpler matrices. Key decompositions: LU (triangular), QR (orthogonal-triangular), eigendecomposition (\( Q\Lambda Q^{-1} \)), SVD (\( U\Sigma V^T \)), and Cholesky (\( LL^T \)).

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

Matrix Decomposition

Factoring a matrix into a product of simpler matrices. Key decompositions: LU (triangular), QR (orthogonal-triangular), eigendecomposition (\( Q\Lambda Q^{-1} \)), SVD (\( U\Sigma V^T \)), and Cholesky (\( LL^T \)).

Why it matters for interviews

Matrix decompositions are the computational backbone of numerical finance: solving linear systems (LU), least squares regression (QR), PCA (eigendecomposition), and Monte Carlo simulation of correlated variables (Cholesky).

Definition and Mathematical Foundation

Factoring a matrix into a product of simpler matrices. Key decompositions: LU (triangular), QR (orthogonal-triangular), eigendecomposition (\( Q\Lambda Q^{-1} \)), SVD (\( U\Sigma V^T \)), and Cholesky (\( LL^T \)).

Application in Quantitative Finance

Matrix decompositions are the computational backbone of numerical finance: solving linear systems (LU), least squares regression (QR), PCA (eigendecomposition), and Monte Carlo simulation of correlated variables (Cholesky).

Related Terms

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Frequently Asked Questions

When do you use Cholesky decomposition in finance?▼
To generate correlated random variables for Monte Carlo simulation. Given covariance matrix \( \Sigma = LL^T \), transform independent standard normals z as \( Lz \) to get correlated samples.
What is the relationship between SVD and eigendecomposition?▼
For a symmetric matrix, SVD and eigendecomposition coincide. For non-symmetric or rectangular matrices, SVD always exists: \( A = U\Sigma V^T \). The singular values are square roots of eigenvalues of \( A^TA \).
Why is numerical stability important for decompositions?▼
Covariance matrices in finance can be ill-conditioned (near-singular). QR decomposition and SVD are numerically stable alternatives to direct inversion, preventing catastrophic error amplification.