# LU Decomposition Calculator

## Find the LU factorization of a matrix step by step

The calculator will find (if possible) the LU decomposition of the given matrix $$$A$$$, i.e. such a lower triangular matrix $$$L$$$ and an upper triangular matrix $$$U$$$ that $$$A=LU$$$, with steps shown.

In case of partial pivoting (permutation of rows is needed), the calculator will also find the permutation matrix $$$P$$$ such that $$$PA=LU$$$.

Related calculator: QR Factorization Calculator

The LU Decomposition Calculator is an online tool for immediate matrix factorization. Our LU solver will help you decompose your matrix quickly and easily.

## How to Use the LU Decomposition Calculator?

### Input

Input the elements of the matrix you wish to decompose into the provided fields.

### Calculation

Click on the "Calculate" button. The LU solver will then process your input and quickly decompose your matrix into a lower triangular matrix $$$L$$$ and an upper triangular matrix $$$U$$$. It will also output a permutation matrix $$$P$$$ if it is different from the identity matrix.

### Result

The $$$L$$$ and $$$U$$$ matrices will be displayed as the output of the calculator. You can then use these matrices for further computations or analysis as per your requirements.

## What Does It Mean to Decompose a Matrix?

Matrix decomposition, also known as matrix factorization, is the process of breaking down a given matrix into a product of simpler matrices. These simpler matrices, often having specific properties, are easier to use for computations such as solving linear equations, finding determinants, or calculating inverses. Common types include LU, eigenvalue, and singular value decompositions. However, not all matrices can be decomposed in all ways (this depends on their characteristics).

## What Is LU Decomposition?

LU decomposition, sometimes referred to as LU factorization, is a strategy in linear algebra that decomposes a matrix into the product of a lower triangular matrix $$$L$$$ and an upper triangular matrix $$$U$$$.

Formally, if $$$A$$$ is a matrix, we can write this as

$$A=LU,$$where:

- $$$A$$$ is the initial matrix
- $$$L$$$ is a lower triangular matrix (all entries above the main diagonal are zero)
- $$$U$$$ is an upper triangular matrix (all entries below the main diagonal are zero)

This process is very helpful for solving linear equations, computing determinants, and finding inverses.

For example, let's take a 2x2 matrix $$$A$$$:

$$A=\left[\begin{array}{cc}3&4\\2&1\end{array}\right]$$The LU decomposition of $$$A$$$ would yield:

$$L=\left[\begin{array}{cc}1&0\\\frac{2}{3}&1\end{array}\right]$$$$U=\left[\begin{array}{cc}3&4\\0&-\frac{5}{3}\end{array}\right]$$So $$$LU$$$ yields the original matrix $$$A$$$. Note that the $$$L$$$ and $$$U$$$ matrices may vary depending on the specific LU decomposition method used (e.g., Doolittle, Crout, or Cholesky).

## Does LU Decomposition Always Exist?

No, LU decomposition does not always exist. It requires that all leading principal minors of the matrix are non-zero. If this condition isn't met, LU decomposition is not possible. However, variations such as LU decomposition with partial pivoting (LUP decomposition) can handle matrices where standard LU decomposition fails. Our calculator performs LUP decomposition as well.

## Why Choose Our LU Decomposition Calculator?

### Ease of Use

With a simple and intuitive interface, inputting your matrix and getting results is as easy as a click of a button.

### Speed and Efficiency

Our LU solver quickly decomposes your matrix, saving you valuable time.

### Accuracy

The calculator provides highly accurate results, ensuring correct LU decompositions.

### Suitable for All

Whether you're a student, teacher, or professional, our calculator is perfect for you.

### FAQ

#### Does every square matrix have an LU decomposition?

No, not every matrix has LU decomposition. LU decomposition is possible only when all leading principal minors of the matrix are nonzero. If this condition isn't met, LU decomposition is not possible. However, variations such as LU decomposition with partial pivoting (LUP decomposition) can handle matrices where standard LU decomposition is not possible.

#### Can I use the LU Decomposition Calculator for matrices?

Our LU Decomposition Calculator is designed to handle matrices of different sizes.

#### Can the LU Decomposition Calculator help me with other matrix operations?

Our LU Decomposition Calculator is specifically designed for LU decompositions. However, if required, the results can be used for further computations, such as solving linear systems or finding determinants.