QR Factorisation Calculator

Category: Linear Algebra
- August 05, 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

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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

Understanding QR Factorization

QR factorization, or QR decomposition, is a way to break down a matrix into two specific types of matrices: an orthogonal matrix (Q) and an upper triangular matrix (R). This means if you take the matrix A, you can express it as A = QR. The orthogonal matrix Q has columns that are at right angles to each Other, while R holds the upper triangular shape where everything below the diagonal is zero. Understanding this process is key to many applications in maths and Science.

The QR Factorization Calculator

The QR Factorization Calculator helps you easily find the QR decomposition of any given matrix. You start by entering your matrix dimensions and values, and the calculator does all the hard work for you. It will show you the matrices Q and R and verify the results. This tool is useful for students and professionals who need quick and accurate matrix calculations without delving deep into complex methods.

How to Use the Calculator

Using the QR Factorization Calculator is straightforward. You first select the number of rows and columns for your matrix. After that, you’ll input the values for the matrix itself. Additionally, you can choose how you want to perform the QR factorization, whether through the Gram-Schmidt process or Householder reflections. You can also decide how many decimal places to display and if you want to see detailed calculation steps.

Key Features of the Calculator

  • Supports multiple matrix sizes: You can work with 2x2, 3x3, or 4x4 matrices.
  • Choose decomposition methods: Pick between the Gram-Schmidt process and Householder reflections.
  • Verification options: Check for orthogonality and multiply Q and R to see if you get A back.
  • Display options: Adjust decimal places and decide if you want to see detailed steps of the calculations.

Applications of QR Factorization

QR factorization has many practical applications across diverse fields. It's often used in solving linear least squares problems, which helps find the best fit for data points. It's also essential for computing eigenvalues in advanced math and can be used in areas like data analysis, signal processing, and machine learning. This versatility makes the QR factorization a valuable tool for anyone working with matrices.

Understanding Orthogonal Matrices

An orthogonal matrix, like Q in the QR factorization, has special properties that make it very useful in calculations. The columns of Q are orthonormal, meaning they are not only perpendicular but also have a unit length. This property allows for effective calculations, especially in numerical methods, as it helps reduce errors that can occur in matrix computations. Furthermore, an orthogonal matrix maintains the length of vectors, which can be crucial in various applications.

The Importance of Verification

Verifying the results of your factorization is crucial for ensuring accuracy. The calculator provides options to check if multiplying Q and R gives you back the original matrix A. It also allows you to confirm the orthogonality of Q by checking if QTQ equals the identity matrix. These checks help you trust your results, making the calculator not just a computational tool but also a learning aid.

Getting the Most Out of Your Calculations

To make the most of the QR Factorization Calculator, consider experimenting with different matrix values and sizes. Try using both decomposition methods to see how the results differ. Additionally, utilise the step-by-step calculation feature to deepen your understanding of the QR process. This hands-on practice can improve your skills in Linear Algebra and matrix manipulation, paving the way for more complex mathematical concepts.