QR Decomposition Calculator

- May 11, 2025
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Calculate the QR decomposition of a matrix, where A = QR with Q being an orthogonal matrix and R being an upper triangular matrix.

This calculator uses the Gram-Schmidt process to find the QR factorisation of any matrix with linearly independent columns.

Matrix Input

Understanding QR Decomposition

QR decomposition is a fundamental matrix factorisation method in linear algebra that decomposes a matrix into an orthogonal matrix Q and an upper triangular matrix R.

Gram-Schmidt Process

The QR decomposition is commonly computed using the Gram-Schmidt process, which follows these steps:

  1. Start with a matrix A with columns a1, a2, ..., an
  2. Compute an orthogonal basis u1, u2, ..., un from the columns of A
  3. Normalise these vectors to create orthonormal vectors q1, q2, ..., qn
  4. The Q matrix consists of these orthonormal columns: Q = [q1 q2 ... qn]
  5. The R matrix contains the projections of original columns onto the orthonormal basis
Mathematical Definition

For an m×n matrix A with linearly independent columns:

  • Q is an m×n matrix with orthonormal columns
  • R is an n×n upper triangular matrix with positive diagonal elements
  • A = QR

The orthogonality of Q means that QTQ = I (identity matrix).

Key Properties
  • If A has linearly independent columns, its QR decomposition is unique
  • For a square matrix A with full rank, Q is orthogonal (QT = Q-1)
  • QR decomposition is numerically stable compared to some other matrix factorizations
  • Computing QR decomposition is generally more efficient than computing the inverse of a matrix

What is QR Decomposition?

QR decomposition is a way to break down a matrix into two special parts: an orthogonal matrix (Q) and an upper triangular matrix (R). This method is useful in various areas of Mathematics and engineering. The orthogonal matrix has columns that are at right angles to each Other, while the upper triangular matrix has all its non-zero entries above the diagonal. By using QR decomposition, you can simplify complex matrix problems.

Introducing the QR Decomposition Calculator

This QR Decomposition Calculator helps you quickly and accurately find the QR factorisation of any matrix. By inputting a matrix with linearly independent columns, this tool will apply the Gram-Schmidt process to generate the Q and R matrices. This feature is particularly handy for students and professionals working with Linear Algebra.

How to Input Your Matrix

To use the calculator, you'll first need to define the size of your matrix. You can enter the number of rows and columns, and then create the matrix. The input section allows you to fill in your matrix values directly or use options to generate a random or identity matrix. This flexibility makes it easier to work with different types of matrices.

Customisation Options for Your Calculation

The calculator comes with several options to tailor your calculations:

  • Choose decimal precision between 2 to 8 decimal places.
  • Decide whether to show step-by-step calculations or just the results.
  • Opt to display values as fractions for better clarity.

These features help create a more personalised experience while working with matrices.

Understanding the Results

After entering your matrix, the calculator provides clear results. You’ll see the original matrix (A), the orthogonal matrix (Q), and the upper triangular matrix (R). Additionally, the calculator verifies the result by showing that A = QR. This gives you confidence that your calculations are correct.

Verification and Orthogonality Checks

Beyond just giving results, the calculator checks essential properties of your matrices. It verifies that the product of Q and its transpose equals the identity matrix. This orthogonality check assures users that the columns of Q are indeed orthonormal, which is fundamental in QR decomposition.

Step-by-Step Solutions Available

If you're keen to learn how to perform QR decomposition, the calculator provides a step-by-step breakdown of the process. You can follow along with the computations, which helps reinforce your understanding of the Gram-Schmidt process and matrix factorisation principles.

Applications of QR Decomposition

QR decomposition is beneficial in many fields, such as:

  • Solving systems of linear equations.
  • Calculating eigenvalues and eigenvectors.
  • Conducting least squares fitting for data analysis.
  • Performing regression analysis.

This versatility makes the QR Decomposition Calculator a valuable tool for students, researchers, and engineers alike.