# QR-Decomposition

## What is QR-Decomposition?

QR-decomposition, also known as QR factorization, is a fundamental matrix decomposition technique in linear algebra. It decomposes a matrix into two constituent matrices, Q (an orthogonal matrix) and R (an upper triangular matrix). QR-decomposition is used to solve linear systems of equations, to compute eigenvalues and eigenvectors of a matrix, and to solve least squares problems.

## Mathematical Definition of QR-Decomposition

Given a matrix A of size m×n with m ≥ n, the QR-decomposition of A is a pair of matrices Q and R such that:

• Q is an m×m orthogonal matrix, meaning that its columns are orthogonal unit vectors (Q

T

Q = I, where I is the identity matrix).

• R is an m×n upper triangular matrix, which means that all the entries below the main diagonal are zero.
• The product of Q and R gives the original matrix A (A = QR).

When A is square (m = n), R is also a square upper triangular matrix, and the factorization is unique if A is invertible and the diagonal elements of R are required to be positive.

## Methods for Computing QR-Decomposition

There are several numerical methods to compute the QR-decomposition of a matrix:

• Gram-Schmidt process: This method orthogonalizes the columns of A by subtracting the projection of each column onto the subspace formed by the previous columns. It is conceptually simple but can be numerically unstable.
• Householder reflections: This approach uses a series of reflection matrices to transform A into an upper triangular matrix. It is more numerically stable than the Gram-Schmidt process and is widely used in numerical computing.
• Givens rotations: Givens rotations use a series of rotational matrices to zero out the sub-diagonal entries of A. This method is particularly useful for sparse matrices.
• Modified Gram-Schmidt process: This is a numerically stable version of the Gram-Schmidt process that re-orthogonalizes the vectors to combat the loss of orthogonality due to rounding errors.

## Applications of QR-Decomposition

QR-decomposition is used in various applications across scientific computing, engineering, and applied mathematics:

• Solving Linear Systems: QR-decomposition can be used to solve the linear system Ax = b by first decomposing A into QR and then solving Rx = QTb, which is simpler as R is an upper triangular matrix.
• Least Squares Problems:

In regression analysis, QR-decomposition is used to solve least squares problems where the goal is to find an approximate solution that minimizes the sum of the squares of the residuals.

• Eigenvalue Computations: The QR algorithm, which iteratively applies QR-decomposition, is a common method for finding the eigenvalues and eigenvectors of a matrix.
• Stability Analysis: In control theory, QR-decomposition is used to analyze the stability of linear systems.