# 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

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If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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 $R$:

#### 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.