Quick answer

Enter a symmetric matrix A, choose a size, and the calculator returns L with A = L Lᵀ when the matrix is valid for Cholesky decomposition.

Formula

  • A = L Lᵀ
  • Supported sizes: 2×2, 3×3, 4×4
  • Formats: integers, decimals, fractions

Introduction

The Cholesky Decomposition Calculator runs entirely in your browser: nothing is uploaded, and results update as soon as the matrix is complete and valid.

Labels follow a₁, b₂ style positions so you can map entries to textbook subscripts. Empty cells above the diagonal in the output are intentional because L is lower triangular.

If you want the math behind the grid, read how to perform Cholesky decomposition and compare manual steps to live output. Worked examples give matrices you can paste in immediately.

What the calculator computes

Internally the tool applies the standard Cholesky update formulas in column order, the same sequence described in the formula article on this site.

Symmetry is checked before factorization. If A[i,j] and A[j,i] disagree beyond tolerance, you see a symmetry error instead of a misleading L.

Positive definiteness is enforced by the algorithm itself. The first nonpositive quantity under a square root stops the run with a clear message.

Results format to three decimal places rounded upward for display consistency. Reconstruction with those rounded values may differ slightly from your exact fraction work.

Fraction input such as 3/4 is parsed to a rational value before computation, which helps when homework uses exact entries.

The calculator is a learning aid and a checker, not a replacement for understanding when Cholesky applies.

Same formulas, automated

  • A = L Lᵀ
  • Column-wise construction of L
  • Errors when symmetry or SPD fails

Knowing the formulas helps you interpret error messages. A symmetry error points to entry typos; a positive definiteness error points to the matrix class.

After a successful run, multiply L Lᵀ on paper with displayed values to see how rounding propagates.

Switching between 2×2, 3×3, and 4×4 clears the grid so sizes do not mix accidentally.

How to use the calculator

  1. Choose matrix size. Pick 2×2, 3×3, or 4×4 before entering numbers.
  2. Enter all cells of A. Use integers, decimals, or fractions like -1/2.
  3. Read L in the result panel. Only the lower triangle is shown; upper positions stay blank.
  4. Fix reported errors. Symmetry errors need matching off-diagonals; SPD errors need a different matrix or method.
  5. Cross-check one example by hand. Do this once per size you use regularly.

Calculator session

Enter A = [[25, 15, -5], [15, 18, 0], [-5, 0, 11]] in 3×3 mode. The tool should return L with first column 5, 3, -1.

If you transpose 15 and -5 by mistake, symmetry checking catches the problem before factorization.

For a failing matrix such as [[1, 2], [2, 1]], expect a positive definiteness message rather than a complex L.