喬列斯基分解計算機

喬列斯基分解計算機

The Cholesky Decomposition Calculator factors a symmetric positive-definite matrix A into the product of a lower-triangular matrix L and its transpose Lᵀ, so that A = LLᵀ. This factorization is fundamental in numerical linear algebra, offering roughly twice the efficiency of general LU decomposition by exploiting the symmetry and positive-definiteness of the input matrix. The calculator provides animated step-by-step derivations, interactive cell highlighting, and automatic verification that LLᵀ reconstructs A.

How Cholesky Decomposition Works

Given an n×n symmetric positive-definite matrix A, the algorithm computes L column by column. For each column j:

Diagonal element:

$$L_{jj} = \sqrt{A_{jj} - \sum_{k=1}^{j-1} L_{jk}^2}$$

Off-diagonal elements (for i > j):

$$L_{ij} = \frac{1}{L_{jj}} \left( A_{ij} - \sum_{k=1}^{j-1} L_{ik} L_{jk} \right)$$

The algorithm proceeds left to right across columns. Each diagonal element involves a square root, which is guaranteed to be real and positive when A is positive-definite. If a negative value appears under the square root, the matrix is not positive-definite.

Conditions for Cholesky Decomposition

Key Properties

How to Use the Cholesky Decomposition Calculator

1. Select matrix size — Choose from 2×2 up to 6×6. Cholesky decomposition requires a square matrix.
2. Enter values — Fill in the matrix cells. The calculator auto-mirrors entries across the diagonal to enforce symmetry (editing A[i,j] automatically sets A[j,i]).
3. Click Decompose — Press the "Decompose A = LLᵀ" button to compute the factorization.
4. Explore the result — Review the color-coded equation A = L × Lᵀ. Click any cell in L to see its derivation formula. Use "Play All" to auto-step through every element.
5. Verify — The calculator multiplies L × Lᵀ back together and reports the maximum error, confirming the decomposition is correct.

Real-World Applications

Cholesky vs Other Decompositions

Frequently Asked Questions

What is Cholesky decomposition?

Cholesky decomposition (named after Andre-Louis Cholesky) factors a symmetric positive-definite matrix A into A = LLᵀ, where L is a lower-triangular matrix with positive diagonal entries. It is one of the most efficient and numerically stable matrix factorizations available.

When can Cholesky decomposition be applied?

The matrix must be symmetric (A = Aᵀ) and positive-definite (all eigenvalues strictly positive, or equivalently, xᵀAx > 0 for every nonzero vector x). Common examples include covariance matrices, correlation matrices, Gram matrices (XᵀX for full-rank X), and stiffness matrices in structural engineering.

What if my matrix is not positive-definite?

If the matrix is not positive-definite, you will encounter a negative value under a square root during the decomposition, which is not a real number. The calculator will report an error indicating exactly which diagonal step failed. You may want to check your matrix for symmetry errors, or consider LDLᵀ decomposition for positive semi-definite matrices.

How is Cholesky decomposition used to solve linear systems?

To solve Ax = b, first decompose A = LLᵀ. Then solve Ly = b by forward substitution (since L is lower-triangular), and then solve Lᵀx = y by back substitution. This is about twice as fast as solving via LU decomposition because L and Lᵀ share the same data.

What is the relationship between Cholesky and the determinant?

Since A = LLᵀ, we have det(A) = det(L) × det(Lᵀ) = det(L)². And since L is triangular, det(L) is simply the product of its diagonal entries. This provides an efficient way to compute the determinant of a positive-definite matrix.

Can Cholesky decomposition be applied to complex matrices?

Yes, for complex matrices the condition is that A must be Hermitian positive-definite (A = A*, where A* is the conjugate transpose). The decomposition becomes A = LLᵀ where Lᵀ is replaced by L* (the conjugate transpose of L). This calculator handles real-valued matrices.