Cholesky Decomposition Calculator

What is Cholesky Decomposition?

The Cholesky decomposition (or Cholesky factorization) is a matrix decomposition that factors a symmetric, positive-definite matrix $A$ into the product of a lower triangular matrix $L$ and its transpose $L^T$:

Requirements for Cholesky Factorization

For a real matrix $A$ to have a unique Cholesky decomposition, it must satisfy two essential properties:

1. Symmetry ($A = A^T$): The matrix must equal its own transpose, meaning $a_{ij} = a_{ji}$ for all row and column indices.
2. Positive-Definiteness ($x^T A x > 0$ for all $x \neq 0$): By Sylvester's criterion, this holds if and only if all leading principal minors (the determinants of top-left submatrices) are strictly greater than zero: $$\det(A_{1 \times 1}) > 0, \quad \det(A_{2 \times 2}) > 0, \quad \det(A_{3 \times 3}) > 0$$

Step-by-Step Cholesky Algorithm Formulas

1. For a $2 \times 2$ Matrix

Given $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{22} \end{pmatrix}$, the lower triangular matrix $L = \begin{pmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{pmatrix}$ is calculated as:

• $l_{11} = \sqrt{a_{11}}$
• $l_{21} = \frac{a_{12}}{l_{11}}$
• $l_{22} = \sqrt{a_{22} - l_{21}^2}$

2. For a $3 \times 3$ Matrix

For $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{pmatrix}$ and $L = \begin{pmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33} \end{pmatrix}$:

• $l_{11} = \sqrt{a_{11}}$
• $l_{21} = \frac{a_{12}}{l_{11}}, \quad l_{31} = \frac{a_{13}}{l_{11}}$
• $l_{22} = \sqrt{a_{22} - l_{21}^2}$
• $l_{32} = \frac{a_{23} - l_{31} l_{21}}{l_{22}}$
• $l_{33} = \sqrt{a_{33} - l_{31}^2 - l_{32}^2}$

$LDL^T$ Decomposition (Square-Root-Free Variant)

The $LDL^T$ decomposition decomposes $A = L D L^T$ where $L$ is a unit lower triangular matrix (diagonal elements equal 1) and $D$ is a diagonal matrix containing the squared diagonal factors. This formulation avoids computing square roots, which is useful in fixed-point embedded computing.

Applications of Cholesky Decomposition

• Solving Linear Systems: For symmetric positive-definite systems $Ax = b$, Cholesky decomposition is roughly twice as fast and much more numerically stable than standard Gaussian elimination or LU decomposition.
• Monte Carlo Simulation: Generates correlated random variables by multiplying uncorrelated standard normal vectors by the Cholesky factor $L$.
• Kalman Filtering: Used in navigation and signal processing to update covariance matrices accurately.
• Machine Learning: Fundamental to Gaussian process regression and kernel methods.

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