What is qr?

The qr function performs QR decomposition of a matrix, breaking it into two components:

• Q: An orthogonal matrix (or unitary matrix in the case of complex numbers).
• R: An upper triangular matrix.

Mathematically, given a matrix A, QR decomposition satisfies:

A = Q * R

QR decomposition is widely used in numerical linear algebra for solving linear systems, eigenvalue problems, and matrix factorization.

Syntax of the qr Function

The syntax for using the qr function is straightforward:

qr(A)

Here, A is the input matrix, which can be real or complex. The function returns an object containing Q and R.

Examples of Using qr

Here are practical examples demonstrating how to perform QR decomposition:

• Example 1: Decomposing a 4x3 matrix.

  
  qr([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
  

  Result:

  • Q = [[0.5, -0.5, 0.5, 0.5], [0.5, 0.5, -0.5, 0.5], [0.5, 0.4999, 0.5, -0.4999], [0.5, -0.5, -0.4999, -0.5]]
  • R = [[1.999, 3, 2], [0, 5, -2], [0, 0, 4], [0, 0, 0]]

Mathematical Context: QR Decomposition

QR decomposition is a cornerstone of numerical methods and linear algebra. Key concepts related to QR decomposition include:

• Orthogonal Matrices: The columns of Q are orthonormal vectors, meaning they are mutually perpendicular and have unit length.
• Upper Triangular Matrix: The matrix R has all zero entries below the main diagonal.
• Stability: QR decomposition provides numerically stable solutions for linear systems and eigenvalue problems.
• Applications: QR decomposition is used in solving least-squares problems, eigenvalue algorithms (like QR iteration), and matrix factorization in scientific computing.

Numerical Importance of QR Decomposition

QR decomposition is preferred over other factorizations like LU in certain numerical problems due to its stability when dealing with nearly singular matrices. It is often used in iterative algorithms where maintaining numerical precision is critical.

Applications of QR Decomposition

QR decomposition has extensive applications in various fields of mathematics, engineering, and computational science:

• Solving Linear Systems: Especially in least-squares problems for overdetermined systems.
• Eigenvalue Computations: Iterative algorithms like QR iteration rely on QR decomposition.
• Data Analysis: Dimensionality reduction techniques such as Principal Component Analysis (PCA) use orthogonal matrices.
• Signal Processing: QR decomposition is a core step in adaptive filtering and beamforming algorithms.

Related Functions

Explore these functions for related operations and further matrix factorizations:

• lup: Computes the LU decomposition of a matrix.
• slu: Computes sparse LU decomposition for large matrices.