# QR Factorization Calculator

## Find the QR decomposition of a matrix step by step

The calculator will find the QR factorization of the given matrix $$$A$$$, i.e. such an orthogonal (or semi-orthogonal) matrix $$$Q$$$ and an upper triangular matrix $$$R$$$ that $$$A=QR$$$, with steps shown.

Related calculator: LU Decomposition Calculator

The QR Factorization Calculator is a resource for efficient and precise matrix factorization. Our digital tool excels in delivering step-by-step QR decomposition.

## How to Use the QR Factorization Calculator?

### Input

Input the elements of the matrix you want to factorize in the provided fields. The matrix can be either square or rectangular.

### Calculation

Click on the "Calculate" button. The QR decomposition calculator will quickly decompose your matrix into an orthogonal matrix $$$Q$$$ and an upper triangular matrix $$$R$$$.

### Result

The calculated matrices $$$Q$$$ and $$$R$$$ will be displayed as the output. You can use these matrices for further computations or analysis as needed.

## What Is QR Factorization?

QR Factorization or QR Decomposition is a technique in linear algebra where a matrix $$$A$$$ is expressed as the product of an orthogonal matrix $$$Q$$$ and an upper triangular matrix $$$R$$$. Mathematically, it is represented as

$$A = QR$$Here,

- $$$Q$$$ is an orthogonal matrix, which means its transpose equals its inverse, i.e., $$$Q^T=Q^{-1}$$$.
- $$$R$$$ is an upper triangular matrix (all entries below the main diagonal are zeros).

This decomposition technique is important in several mathematical operations, such as solving linear equations, matrix inversion, and calculating eigenvalues.

For example, consider a 3x3 matrix $$$A$$$:

$$A=\left[\begin{array}{ccc}12&-51&4\\6&167&-68\\-4&24&-41\end{array}\right]$$After performing QR factorization, we will obtain the following matrices $$$Q$$$ and $$$R$$$:

$$Q=\left[\begin{array}{ccc}\frac{6}{7}&-\frac{69}{175}&-\frac{58}{175}\\\frac{3}{7}&\frac{158}{175}&\frac{6}{175}\\-\frac{2}{7}&\frac{6}{35}&-\frac{33}{35}\end{array}\right]$$$$R=\left[\begin{array}{ccc}14&21&-14\\0&175&-70\\0&0&35\end{array}\right]$$It can be checked that, indeed, $$$A=QR$$$.

## Is QR factorization unique?

The QR factorization of a matrix is not always unique. Different algorithms may yield different results. However, there is a unique "reduced" or "thin" QR factorization where the diagonal elements of $$$R$$$ are all positive. Without such constraints, QR factorization is not unique.

## Why Choose Our QR Factorization Calculator?

### Efficiency

Our calculator provides fast and accurate QR factorization, saving you time and effort.

### Step-by-Step Solutions

Besides delivering the final answer, our calculator also provides a step-by-step solution, helping you understand the process better.

### Accessibility

The calculator is online-based and can be accessed anytime, anywhere from your device.

### Ease of Use

The user-friendly interface makes it simple to input your matrix and obtain the answer.

### FAQ

#### Does every matrix have a QR decomposition?

Yes, every square matrix and even rectangular matrix has a QR decomposition.

#### What is the QR Factorization Calculator?

The QR Factorization Calculator is a free online tool that decomposes a matrix into the product of an orthogonal matrix $$$Q$$$ and an upper triangular matrix $$$R$$$.

#### Can I use the QR Factorization Calculator for non-square matrices?

Yes, our QR Factorization Calculator can handle both square and rectangular matrices.

#### How accurate is the QR Factorization Calculator?

Our QR Factorization Calculator is designed to provide highly accurate results, making it a reliable tool for matrix factorization.