Calculator Form
Example Data Table
This sample symmetric positive definite matrix produces a valid lower triangular Cholesky factor.
| Example item | Value | Notes |
|---|---|---|
| Matrix A | [[25, 15, -5], [15, 18, 0], [-5, 0, 11]] | Symmetric positive definite |
| Lower factor L | [[5, 0, 0], [3, 3, 0], [-1, 1, 3]] | A = L × LT |
| Determinant | 2025 | (5 × 3 × 3)2 |
| Example vector b | [35, 33, 6] | Used for solving Ax = b |
| Solution x | [1, 1, 1] | Computed using forward and backward substitution |
Formula Used
For a symmetric positive definite matrix A, the decomposition is:
Each entry of the lower triangular matrix L is computed with these formulas:
Lij = ( Aij − Σk=1j−1 LikLjk ) / Ljj, for i > j
When solving Ax = b, first solve Ly = b, then solve LTx = y. The determinant follows this rule:
How to Use This Calculator
- Select the matrix size from 2 × 2 up to 6 × 6.
- Enter a symmetric positive definite matrix in the input boxes.
- Choose the number of decimal places for displayed values.
- Keep system solving enabled if you also want Ax = b solved.
- Enter the right-hand side vector values when needed.
- Use the example button if you want a tested sample matrix.
- Click Compute decomposition to place results above the form.
- Review the lower factor, determinant, reconstruction accuracy, graph, and downloads.
Frequently Asked Questions
1. What is Cholesky decomposition?
Cholesky decomposition factorizes a symmetric positive definite matrix into a lower triangular matrix and its transpose. It is efficient for solving systems, computing determinants, and checking numerical stability.
2. Why must the matrix be symmetric?
The decomposition is derived for symmetric matrices because the factorization form A = L LT assumes mirrored off-diagonal entries. If symmetry fails, the method is not valid in its standard form.
3. What does positive definite mean here?
Positive definite means every nonzero vector produces xTAx > 0. Practically, the diagonal pivots stay positive during factorization, allowing each square root step to remain real and valid.
4. How is this different from LU decomposition?
LU decomposition works for many square matrices, while Cholesky is specialized for symmetric positive definite matrices. Because of that structure, Cholesky needs fewer operations and usually gives better efficiency.
5. Can this calculator solve linear systems too?
Yes. When the solve option is enabled, it computes Ly = b first and then LTx = y. That gives the solution vector x using two stable triangular solves.
6. Why is the determinant easy after decomposition?
The determinant of A equals the square of the product of diagonal entries of L. That is much easier than expanding a determinant directly, especially for larger matrices.
7. What does reconstruction error show?
Reconstruction error measures the largest absolute difference between the original matrix A and the rebuilt matrix L LT. Smaller values indicate strong numerical agreement and a reliable factorization.
8. What matrix size works best here?
This page supports sizes from 2 × 2 to 6 × 6 for clarity and quick entry. It is ideal for learning, checking homework, validating examples, and testing smaller numerical problems.