QR Factorization Calculator

Category: Linear Algebra
- May 14, 2025
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Calculate the QR decomposition of a matrix. This calculator decomposes a matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R.

Matrix Dimensions

Matrix Input

Display Options

About QR Factorization

QR factorization (also called QR decomposition) is a process in linear algebra where a matrix A is decomposed into the product of an orthogonal matrix Q and an upper triangular matrix R (A = QR).

Key Properties
  • Orthogonal matrix Q: Has orthonormal columns, meaning QTQ = I (identity matrix).
  • Upper triangular matrix R: All elements below the main diagonal are zero.
  • Matrix product: A = QR
QR Decomposition Methods
  1. Gram-Schmidt Process: Converts a set of linearly independent vectors into orthogonal vectors.
  2. Householder Reflections: Uses reflections to transform the matrix into upper triangular form.
  3. Givens Rotations: Uses rotations to introduce zeros below the diagonal.
Applications of QR Factorization
  • Solving linear least squares problems
  • Computing eigenvalues (QR algorithm)
  • Solving linear systems of equations
  • Signal processing and data compression
  • Machine learning and statistical analysis

QR Factorization Formula:

A = Q × R

where:

  • A is the original matrix
  • Q is an orthogonal matrix (QTQ = I)
  • R is an upper triangular matrix

What is the QR Factorization Calculator?

The QR Factorization Calculator is an easy-to-use online tool that breaks down a given matrix into two special matrices: an orthogonal matrix (Q) and an upper triangular matrix (R). This decomposition is helpful for solving many types of mathematical problems in algebra, engineering, and data Science. Whether you are studying Linear Algebra, working on a project, or solving equations, this calculator provides quick and accurate QR factorizations without manual calculations.

How to Use the QR Factorization Calculator

  • Step 1: Choose the number of rows and columns for your matrix.
  • Step 2: Enter the values of the matrix in the input fields.
  • Step 3: Select the method for factorization: Gram-Schmidt Process or Householder Reflections.
  • Step 4: Choose how many decimal places you want for your results.
  • Step 5: Click the "Calculate QR Factorization" button to get your results.

You can also opt to show detailed calculation steps and verify the orthogonality of the Q matrix, making learning and reviewing much easier.

Why Use This Calculator?

The QR Factorization Calculator is a valuable resource for anyone who needs to:

  • Understand and apply matrix decomposition techniques.
  • Solve linear least squares problems quickly.
  • Prepare for exams or complete assignments efficiently.
  • Work in areas like signal processing, machine learning, or data compression.

It saves time and minimizes errors that often occur in manual calculations. It's especially useful when paired with Other calculators like the inverse function tool to find inverse functions, or when needing to solve for inverses in complex scenarios.

Benefits of QR Factorization

  • Orthogonal Matrix (Q): Helps maintain numerical stability during calculations.
  • Upper Triangular Matrix (R): Makes solving systems of equations faster and more straightforward.
  • Verification: Confirms results by checking if Q × R reconstructs the original matrix A.
  • Learning Aid: Shows step-by-step breakdowns for better understanding, similar to how an equation solver tool or partial fraction solver would operate for other types of problems.

Frequently Asked Questions (FAQ)

What methods does the calculator use?

You can choose between the Gram-Schmidt Process and Householder Reflections. Both methods ensure accurate decomposition of your matrix.

When should I use QR Factorization?

Use QR Factorization when solving linear systems, least squares problems, or finding eigenvalues. It is often applied alongside techniques like polynomial root finding with a polynomial root finder or simplifying expressions with an expression simplification tool.

What matrix sizes are supported?

The calculator supports matrices with dimensions from 2×2 up to 4×4, ideal for many academic and practical applications.

Is it possible to check if Q is orthogonal?

Yes! The "Verify Orthogonality" option checks if QT × Q equals the identity matrix, ensuring Q is truly orthogonal.

Can I see how the QR Factorization is calculated?

Absolutely. Enable "Show Calculation Steps" to view detailed steps, helping you learn the process just like using a solve for inverses or logarithm equation helper.

Final Notes

The QR Factorization Calculator is part of a growing set of helpful tools for Math learners and professionals, like the midpoint formula tool for coordinate calculations or the complex arithmetic tool for imaginary numbers. Its clear steps, immediate results, and accuracy make it a great choice whether you are trying to solve equations, calculate midpoint values, or even handle advanced concepts with an inverse hyperbolic sine or asinh function tool.