Cholesky Decomposition Calculator

Decompose a symmetric positive-definite matrix into A = L·Lᵀ. Enter the six independent entries of a 3×3 matrix to get the lower triangular factor L, its transpose, and the determinant.

Quick Facts

Method
A = L · Lᵀ (Cholesky-Banachiewicz algorithm)
L is lower triangular with positive diagonal entries; it exists and is unique whenever A is symmetric and positive definite.
Efficiency
About n³/3 arithmetic operations
Roughly half the work of general LU decomposition, which is why it is the standard choice for symmetric positive-definite systems.

Your Results

Calculated
Lower triangular L
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Rows of L, top to bottom
Transpose Lᵀ
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Upper triangular; A = L · Lᵀ
Determinant of A
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det(A) = det(L)²
Matrix classification
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Required for a real Cholesky factor

Ready

Enter the six independent entries of a symmetric matrix, then press Calculate.

How Cholesky Decomposition works

Cholesky decomposition factors a symmetric, positive-definite matrix A into the product of a lower triangular matrix L and its transpose: A = L · Lᵀ. Every diagonal entry of L is positive, and this factorization exists and is unique for exactly the matrices that are symmetric with every leading principal minor positive — the definition of "positive definite."

Formula and method

The Cholesky-Banachiewicz algorithm builds L one entry at a time from the symmetric matrix A = (aᵢⱼ), using only entries already computed:

  • Diagonal entries: Lᵢᵢ = √( aᵢᵢ − Σ (Lᵢₖ)² for k < i )
  • Below-diagonal entries: Lᵢⱼ = ( aᵢⱼ − Σ Lᵢₖ·Lⱼₖ for k < j ) / Lⱼⱼ, for i > j

For the 3×3 case used here, this reduces to six explicit steps: L11 = √A11, L21 = A12/L11, L31 = A13/L11, L22 = √(A22 − L21²), L32 = (A23 − L31·L21)/L22, and L33 = √(A33 − L31² − L32²). If any quantity under a square root is zero or negative, A is not positive definite and no real Cholesky factor exists.

Common sources of error

  • Non-symmetric input: this calculator only asks for the six independent entries and assumes A12 = A21, A13 = A31, and A23 = A32 — Cholesky decomposition is undefined for a matrix that is not symmetric.
  • Not positive definite: a negative or zero value under a square root — often traced to a small or negative diagonal entry — means the matrix fails the positive-definite requirement.
  • Rounding intermediate steps: because each row of L depends on the rows above it, rounding early compounds error — carry full precision through all six steps.

Checking your result

Multiply L by its transpose (L · Lᵀ) and confirm you recover the original matrix A. As a fast cross-check, the product of the diagonal entries of L, squared, equals the determinant of A: det(A) = (L11 × L22 × L33)². Also confirm every diagonal entry of L is positive — a zero, negative, or complex diagonal entry signals invalid input.

Applications

Cholesky decomposition solves symmetric linear systems and least-squares normal equations faster than general-purpose methods, generates correlated random variables in Monte Carlo simulation, and underlies efficient determinant and matrix-inversion routines used throughout statistics, finance, and engineering software.

Frequently Asked Questions

What is Cholesky decomposition?
Cholesky decomposition factors a symmetric positive-definite matrix A into A = L · Lᵀ, where L is a lower triangular matrix with positive diagonal entries. It is computed row by row with the Cholesky-Banachiewicz algorithm, using square roots and previously solved entries of L.
What conditions must the matrix satisfy?
The matrix must be symmetric (A equals its transpose) and positive definite, meaning every leading principal minor has a positive determinant. If any value under a square root becomes zero or negative during the algorithm, the matrix is not positive definite and no real Cholesky factor exists.
How is Cholesky decomposition different from LU decomposition?
LU decomposition factors any square matrix into a lower and an upper triangular matrix and generally needs row pivoting for numerical stability. Cholesky decomposition only applies to symmetric positive-definite matrices, but it exploits that symmetry to use about half the arithmetic of LU decomposition and needs no pivoting.
What is Cholesky decomposition used for?
It is used to solve symmetric linear systems and least-squares normal equations efficiently, to generate correlated random variables in Monte Carlo simulation, and to compute determinants and matrix inverses in statistics, finance, and engineering software.

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