LDL Decomposition Calculator

LDL Decomposition Formula:

\[ A = LDL^T \]

where:

\( A \) is a symmetric matrix
\( L \) is lower triangular with 1s on diagonal
\( D \) is a diagonal matrix

Result:

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1. What is LDL Decomposition?

LDL decomposition (or LDLT decomposition) is a matrix factorization that expresses a symmetric matrix A as the product of a lower triangular matrix L, a diagonal matrix D, and the transpose of L. It's a variant of Cholesky decomposition that works for positive semi-definite matrices.

2. How Does the Calculator Work?

The calculator performs the following decomposition:

\[ A = LDL^T \]

Where:

\( A \) — Input symmetric matrix
\( L \) — Lower triangular matrix with 1s on the diagonal
\( D \) — Diagonal matrix

Algorithm: The calculator implements the standard LDL decomposition algorithm that computes L and D through a series of row operations.

3. Importance of LDL Decomposition

Applications: LDL decomposition is used in numerical linear algebra, optimization problems, solving systems of linear equations, and in the implementation of various statistical methods.

4. Using the Calculator

Tips: Enter your symmetric matrix using commas to separate values within a row and semicolons to separate rows. The matrix must be square and symmetric.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between LDL and Cholesky decomposition?
A: LDL works for positive semi-definite matrices, while Cholesky requires positive definite matrices. LDL avoids square roots in the computation.

Q2: When does LDL decomposition fail?
A: It fails when the matrix is not symmetric or when it has zero elements on the diagonal of D (for matrices that are not positive semi-definite).

Q3: What are the computational benefits of LDL?
A: LDL decomposition has similar computational complexity to Cholesky (O(n³)) but is more numerically stable for certain matrices.

Q4: Can LDL decomposition be used for non-symmetric matrices?
A: No, the decomposition only works for symmetric matrices. For general matrices, other decompositions like LU are more appropriate.

Q5: How is LDL decomposition used in solving linear systems?
A: After decomposing A = LDLᵀ, the system Ax = b can be solved by forward and backward substitution using L and D.