Mathematical Calculators

QR Decomposition Calculator

Find out orthonormal matrix and upper triangular matrix easily with our free online QR decomposition calculator!

QR decomposition calculator

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Table of contents

What is a QR decomposition?
How to calculate QR decomposition?
What is the Gram–Schmidt process?
How does Gram–Schmidt calculation work?
Does QR decomposition always exist?
Where QR factorization is used?
In linear algebra, factorizing a complex matrix makes it easier to analyze. QR decomposition is a matrix decomposition, which commonly used to solve linear systems, obtain eigenvalues, and calculations related to determinants. QR decomposition is also used in machine learning and on its applications.
Our QR decomposition calculator will calculate the upper triangular matrix and orthogonal matrix from the given matrix.
To use our calculator:
1. Add your matrix size (Columns <= Rows)
2. Insert matrix points
3. Choose rounding precision
4. See results
On this page, you will also learn how to calculate QR decomposition with Gram–Schmidt process, and where QR composition is used in real life.

What is a QR decomposition?

QR decomposition is a technique used to convert a matrix into the form A = QR, where R equals upper triangular matrix, Q equals orthogonal matrix, and Q^(T)Q=I holds, where Q^(T) is the Qs' transpose, and I is the matrixs' identity.
QR decomposition is also known as QR factorization and QU factorization, and it is commonly used in solving equations linear systems.
Mathematical defition of QR decomposition

How to calculate QR decomposition?

A QR decomposition can be performed through various methods. These include the Gram–Schmidt process, the Householder transformations, and the Givens rotations.
We will go through Gram–Schmidt process, and here is a step-by-step guide on how to calculate QR decomposition with it:
A = QR,
A = Given matrix
Q = Orthogonal matrix
R = Upper triangular matrix
1. Define matrix A
2. Take columns of A, and process them through Gram–Schmidt process. As of result, you get orthonormal vectors: e1, e2, ..., en.
3. Form a matrix Q with these vectors, by using vectors as columns.
4. Form matrix R by left-multiplying A with the transpose of Q (R = QᵀA)
There you go! You successfully calculate QR decomposition, and founded both orthogonal matrix and upper triangular matrix!
QR Decomposition with Gram-Schmidt method

What is the Gram–Schmidt process?

The Gram-Schmidt process is a sequence of operations designed to transform a set of linearly independent vectors into an equivalent set of orthonormal vectors.

How does Gram–Schmidt calculation work?

The Gram–Schmidt calculation is a mathematical tool used to determine the optimal fit between two sets of data. It’s often used in machine learning and data analysis, and it can be helpful when trying to find the best algorithms or models for predicting outcomes. In short, the Gram–Schmidt algorithm takes two sets of data—say, texts from a training set and predictions made from a model based on that data—and creates a similarity score between them. The higher the score, the more similar the sets are.
The Gram-Schmidt process is commonly used because it processes the calculations in an orthonormal base, which is often a much easier base to perform calculations.
Gram–Schmidt method

Does QR decomposition always exist?

The factorization A = QR decomposition of a matrix A is a useful technique for estimating eigenvalues. It always exists when the rank of A is equal to the number of columns of A.

Where QR factorization is used?

The concept of QR factorization is a very useful framework for various statistical and data analysis applications. One of these is the solution to the least square problems.
QR factorization is also a commonly used component in machine learning and its applications. It can be used for example to automatically remove an object from an image. Another example is extracting an image from a video clip.
QR factorization in data science


Gander, W., 1980. Algorithms for the QR decomposition. Res. Rep, 80(02), pp.1251-1268.
Goodall, C.R., 1993. 13 Computation using the QR decomposition.
Angelica Miller
Article author
Angelica Miller
Angelica is a psychology student and a content writer. She loves nature and wathing documentaries and educational YouTube videos.
QR Decomposition Calculator English
Published: Thu Oct 07 2021
In category Mathematical calculators
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