Cholesky Factorization

Learn how to implement Cholesky factorization of matrices using R, Rccp, Armadillo, and Eigen.

Cholesky decomposition

Cholesky decomposition or factorization is a decomposition of a symmetric and positive definite real matrix into the product of a lower triangular matrix and its conjugate transpose. This is useful for efficient numerical solutions. Cholesky decomposition is roughly twice as efficient as the LU decomposition used for solving systems of linear equations.

A square matrix 𝐴𝐴 defined over the real numbers is defined as symmetric when 𝐴=𝐴𝑇𝐴 = 𝐴^𝑇.

A symmetric (𝑛×𝑛)(𝑛 \times 𝑛) real matrix, 𝑀𝑀, is said to be positive definite if the scalar 𝑧𝑇𝑀𝑧𝑧^𝑇𝑀𝑧 is strictly positive for every nonzero column vector 𝑧𝑧 of 𝑛𝑛 real numbers.

The Cholesky decomposition of a square symmetric positive definite matrix 𝐴𝐴 is a decomposition of the form:


In this equation:

  • 𝐿𝐿 is a lower triangular matrix with real and positive diagonal entries.
  • 𝐿𝑇𝐿^𝑇 denotes the transpose of 𝐿𝐿.

Every real-valued symmetric positive-definite matrix has a unique Cholesky decomposition.

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