QR Decomposition Calculator

Category: Linear Algebra
- April 18, 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 factorization of any matrix with linearly independent columns.

Matrix Input

Understanding QR Decomposition

QR decomposition is a fundamental matrix factorization 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. Normalize 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
QR Decomposition 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 Decomposition Calculator?

The QR Decomposition Calculator helps you break down a matrix into two specific components: an orthogonal matrix (Q) and an upper triangular matrix (R). This process is useful in many areas of linear algebra, especially when solving systems of equations or performing regression analysis.

This tool uses the Gram-Schmidt process to perform the calculation. It’s accurate, fast, and does all the heavy lifting for you, even displaying optional step-by-step explanations. Whether you’re studying or working with real data, this calculator offers a clear path to QR matrix factorization.

Why Use QR Decomposition?

QR decomposition is a widely used matrix factorization technique in numerical analysis and linear algebra. It’s particularly helpful for:

  • Solving linear systems efficiently
  • Handling least squares problems
  • Computing eigenvalues as part of the matrix transformation process
  • Making matrices easier to work with in data analysis or machine learning

How to Use the Calculator

Using the QR Decomposition Calculator is simple:

  1. Enter the number of rows and columns for your matrix.
  2. Click “Create Matrix” to generate input fields.
  3. Fill in your matrix values manually or use the “Random Matrix” or “Identity Matrix” options.
  4. Choose your display preferences, such as decimal precision or showing fractions.
  5. Click “Calculate QR Decomposition” to get the results.

The calculator will show:

  • The original matrix (A)
  • The orthogonal matrix (Q)
  • The upper triangular matrix (R)
  • Verification that A = QR
  • Confirmation that Q is orthogonal (QTQ = I)
  • Optional step-by-step breakdown of the process

Where Else Is QR Used?

This calculator is part of a broader set of matrix tools often used together in linear algebra studies and applications:

FAQs

What is QR Decomposition used for?

It simplifies matrix equations and is essential for solving systems of linear equations, performing data fitting, and in eigenvalue computations.

What kind of matrices can be decomposed?

Any matrix with linearly independent columns can be decomposed using this tool. The number of rows must be greater than or equal to the number of columns.

Is the process accurate?

Yes. The calculator verifies both the result A = QR and the orthogonality of Q using matrix multiplication, ensuring numerical precision.

Do I need to understand the math behind it?

No. The tool provides both the results and an optional step-by-step explanation if you want to learn more.

Can I see intermediate steps?

Yes, just check the box labeled “Show calculation steps” before computing. This is great for learning or verifying your own work.

Conclusion

The QR Decomposition Calculator is a helpful and user-friendly way to analyze and break down matrices for mathematical, academic, or practical applications. Whether you’re exploring matrix decomposition methods or need a quick QR factorization tool, this calculator gives you reliable results with minimal effort.

It complements other powerful matrix tools like the LU matrix factorization tool, matrix inverse tool, pseudoinverse matrix solver, and matrix diagonalization tools — giving you a complete set of resources to work with linear systems and advanced matrix operations.