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Cholesky decomposition factors a symmetric, positive-definite matrix A into the product of a lower triangular matrix L and its transpose: A = L·Lᵀ. It's roughly twice as efficient as general LU decomposition for matrices that meet those conditions, which is why it's the standard choice for solving systems built from covariance matrices, least-squares normal equations, and similar symmetric positive-definite problems.
This calculator computes L for any symmetric positive-definite matrix up to 6×6, showing each diagonal and off-diagonal entry as it's derived.
How it works
Enter a symmetric matrix (a_ij must equal a_ji). The tool works down the diagonal, computing each entry of L from the entries of A and the previously-computed entries of L, and reports an error if a diagonal entry would require a square root of a negative number (meaning A isn't positive-definite).
- Click Calculate to see your results.
Examples
A textbook 3×3 example
For A = [[4,12,-16],[12,37,-43],[-16,-43,98]], Cholesky decomposition gives L = [[2,0,0],[6,1,0],[-8,5,3]], and you can verify L·Lᵀ reconstructs A exactly.
Who should use it
- Solving linear systems arising from least-squares normal equations or covariance matrices.
- Verifying whether a symmetric matrix is positive-definite (the decomposition fails cleanly if it isn't).
Industry applications
- Statistics and machine learning (covariance/precision matrices)
- Numerical linear algebra and scientific computing
Advantages
- About twice as computationally efficient as general LU decomposition for matrices where it applies.
- A very common building block for solving symmetric positive-definite linear systems.
Limitations
- Only works for symmetric, positive-definite matrices — not a general-purpose factorization.
Common mistakes to avoid
- Trying to apply Cholesky decomposition to a non-symmetric matrix, which the method fundamentally doesn't support.
- Assuming any symmetric matrix qualifies — it must also be positive-definite (all its eigenvalues positive) for a real Cholesky factorization to exist.
Best practices
- Use Cholesky decomposition specifically when you know your matrix is symmetric positive-definite (like a covariance matrix) — it's faster and more numerically stable than general LU decomposition for that case.
Tips
- If Cholesky decomposition fails on a matrix you expected to be positive-definite, double check your matrix is exactly symmetric first — small transcription errors are a common cause.