Skip to main content

TL;DR

For a positive definite matrix \( \Sigma \), the unique factorization \( \Sigma = LL^T \) where L is lower triangular with positive diagonal entries. It is the matrix analog of taking a square root.

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

Cholesky Decomposition

For a positive definite matrix \( \Sigma \), the unique factorization \( \Sigma = LL^T \) where L is lower triangular with positive diagonal entries. It is the matrix analog of taking a square root.

Why it matters for interviews

The standard method for generating correlated random variables in Monte Carlo simulation: given independent standard normals z, the vector Lz has covariance \( \Sigma \). It is also used for efficient linear system solving.

Definition and Mathematical Foundation

For a positive definite matrix \( \Sigma \), the unique factorization \( \Sigma = LL^T \) where L is lower triangular with positive diagonal entries. It is the matrix analog of taking a square root.

Application in Quantitative Finance

The standard method for generating correlated random variables in Monte Carlo simulation: given independent standard normals z, the vector Lz has covariance \( \Sigma \). It is also used for efficient linear system solving.

Related Terms

Ready to practice for the Quant Trading Interview?

Adaptive practice powered by Item Response Theory targets your weak areas. Start with 3 free sessions.

Start free practice →

Frequently Asked Questions

How is Cholesky used in Monte Carlo for correlated assets?▼
To simulate n correlated assets: (1) estimate covariance matrix \( \Sigma \), (2) compute \( L \) via Cholesky, (3) generate independent \( z \sim N(0,I) \), (4) set correlated sample \( x = \mu + Lz \). Then x has the desired correlation structure.
Why is Cholesky preferred over eigendecomposition for this?▼
Cholesky is about twice as fast (\( O(n^3/3) \) vs \( O(n^3) \)), numerically stable, and produces the same result for positive definite matrices. It is the default choice when only the 'square root' is needed.
What happens if the matrix is not positive definite?▼
Cholesky fails -- the algorithm encounters a negative value where a square root is needed. This signals that the covariance matrix estimate is degenerate, typically because n < p (more assets than observations). Regularization is needed.