What is QR Decomposition?

Decomposition or Factorization is dividing the original single entity into multiple entities for easiness. Decomposition has various applications in numerical linear algebra, optimization, solving systems of linear equations, etc. QR decomposition is a versatile tool in numerical linear algebra that finds applications in solving linear systems, least squares problems, eigenvalue computations, etc. Its numerical stability and efficiency make it a valuable technique in a range of applications.

QR decomposition, also known as QR factorization, is a fundamental matrix decomposition technique in linear algebra. QR decomposition is a matrix factorization technique that decomposes a matrix into the product of an orthogonal matrix (Q) and an upper triangular matrix (R). Given a matrix A (m x n), where m is the number of rows and n is the number of columns, the QR decomposition can be expressed as:

QR decomposition finds widespread use in machine learning for tasks like solving linear regression, eigenvalue problems, Gram-Schmidt orthogonalization, handling overdetermined systems, matrix inversion, Gram matrix factorization, and enhancing numerical stability in various algorithms. More details about it, is in the application section.

QR Decomposition Related Concepts

1. Matrix Factorization: Matrix factorization involves expressing a matrix as the product of two or more matrices. In QR decomposition, we express a given matrix A as the product of an orthogonal matrix Q and an upper triangular matrix R.
2. Orthogonal Matrix: An orthogonal matrix Q has the property that its transpose is equal to its inverse (Q^T * Q = I, where I is the identity matrix).
   • Properties: Orthogonal matrices preserve the length of vectors and the dot product. They play a crucial role in QR decomposition.
3. Upper Triangular Matrix: A matrix is upper triangular if all entries below the main diagonal are zero. In QR decomposition, R is an upper triangular matrix.
4. Gram-Schmidt Process (Orthogonalization Process): The Gram-Schmidt process is used to orthogonalize a set of vectors. In the context of QR decomposition, it is applied to the columns of the original matrix to construct an orthogonal matrix Q.

QR decomposition is a way of expressing a matrix as the product of two matrices: Q (an orthogonal matrix) and R (an upper triangular matrix). In this article, I will explain decomposition in Linear Algebra, particularly QR decomposition among many decompositions.