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 the QR Decomposition Calculator?

The QR Decomposition Calculator is a powerful tool designed to help users decompose a matrix into two specific components: an orthogonal matrix (Q) and an upper triangular matrix (R). This decomposition is vital in various mathematical applications, especially in Linear Algebra. By inputting a matrix with linearly independent columns, users can quickly obtain the QR factorization, which is useful for solving equations and performing Other complex calculations.

How Does It Work?

The calculator employs the Gram-Schmidt process to perform the QR decomposition. This method systematically transforms the original matrix into an orthogonal basis, enabling users to find the Q and R matrices efficiently. Once the decomposition is achieved, the calculator displays the original matrix, the orthogonal matrix, and the upper triangular matrix, providing an easy overview of the results.

Benefits of Using the QR Decomposition Calculator

Using this calculator comes with several advantages:

  • Reduces complex calculations into easy steps.
  • Visualizes the results, making it easier to understand the decomposition process.
  • Allows adjustments for decimal precision and optional display formats, such as fractions.
  • Provides step-by-step solutions to enhance learning and comprehension.

Inputting Your Matrix

To use the QR Decomposition Calculator, users need to define the matrix dimensions first. The calculator allows for matrices with a range of row and column sizes, making it flexible for different needs. Once the dimensions are set, users can input their matrix values directly into the provided fields.

Advanced Options for Customisation

The calculator also offers advanced options to tailor the output to individual preferences. Users can choose the number of decimal places for the results, decide whether to display calculation steps, and even select if they prefer fractions over decimals. These options provide a more user-friendly experience, accommodating various mathematical styles.

Results You Can Trust

Once the calculation is complete, the calculator presents the results clearly. You will see the original matrix, the orthogonal matrix (Q), and the upper triangular matrix (R). There's also a verification section that checks if the product of Q and R returns to the original matrix (A), ensuring accuracy in the results.

The Importance of QR Decomposition

QR decomposition plays a crucial role in several applications across Mathematics and engineering. Its uses include solving systems of linear equations, computing eigenvalues, and fitting data in regression analysis. By understanding these applications, users can appreciate why QR decomposition is such an essential tool in advanced mathematics.

Learning More About the Gram-Schmidt Process

For those interested in the underlying methods, the QR Decomposition Calculator is also a gateway to learning the Gram-Schmidt process. This method not only helps in understanding QR decomposition but also strengthens foundational knowledge in linear algebra. By mastering this process, users can tackle more complex problems confidently.