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 * L^{T}

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.

- First, ensure the matrix is symmetric and positive-definite.
- Decompose the matrix A into L * L
^{T}, where L is a lower triangular matrix. - Compute the entries of L by solving the resulting triangular system.
- Verify the factorization by multiplying L and L
^{T}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 x^{T}Ax 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, itâ€™s best to consult mathematical software or a numerical analyst.