Enter your matrix size and entries into the calculator to determine its Cholesky factorization.

Cholesky Factorization Formula

The following formula is used to calculate the Cholesky factorization of a matrix.

A = L * LT

Variables:

  • A is the original symmetric, positive-definite matrix
  • L is the lower triangular matrix with real and positive diagonal entries

To calculate the Cholesky factorization, the matrix A is decomposed into a product of a lower triangular matrix L and its transpose.

What is Cholesky Factorization?

Cholesky factorization is a numerical method used to solve linear algebra problems, particularly those involving symmetric, positive-definite matrices. It decomposes a matrix into a product of a lower triangular matrix and its transpose, simplifying the process of solving systems of linear equations, inverting matrices, and computing determinants.

How to Calculate Cholesky Factorization?

The following steps outline how to calculate the Cholesky factorization using the given formula.

  1. First, ensure the matrix is symmetric and positive-definite.
  2. Decompose the matrix A into L * LT, where L is a lower triangular matrix.
  3. Compute the entries of L by solving the resulting triangular system.
  4. Verify the factorization by multiplying L and LT to check if the original matrix A is obtained.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Matrix A:

[ 4, 12, -16 ]
[ 12, 37, -43 ]
[ -16, -43, 98 ]

FAQ

1. What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose. In other words, the elements across the main diagonal are mirror images.

2. What does positive-definite mean?

A matrix is positive-definite if it has all positive eigenvalues and the quadratic form xTAx is positive for all non-zero vectors x.

3. Why use Cholesky factorization?

Cholesky factorization is computationally efficient and numerically stable, making it ideal for solving large linear systems and other applications in numerical analysis.

4. Can Cholesky factorization be used for all matrices?

No, it is specifically for symmetric, positive-definite matrices. Other factorizations like LU or QR might be used for general matrices.

5. Is the calculator accurate?

The calculator provides an estimate of the Cholesky factorization based on the inputs provided. For exact figures, it’s best to consult mathematical software or a numerical analyst.