Singular Value Decomposition (SVD) is a fundamental matrix factorization technique in linear algebra with wide - ranging applications in various fields such as data analysis, computer vision, and signal processing. Let's calculate the Singular Value Decomposition (SVD) of a simple (2 x 2) matrix A
Definition and Basic Concepts: Given a matrix A of size (m x n), the Singular Value Decomposition of A is expressed as A = UΣV(Τ), where U is an (m x m) orthogonal matrix, meaning (U(T)*U = U*U(T), where I is the identity matrix. The columns of U are called the left - singular vectors of A. Σ is an (m x n) diagonal matrix with non - negative real numbers on the diagonal. These diagonal elements are called the singular values of A and are usually arranged in descending order. V is an (n x n) orthogonal matrix, and its columns are the right - singular vectors of A. V(T) represents the transpose of V Geometric Interpretation SVD can be thought of as a transformation that decomposes a linear transformation represented by the matrix A into a sequence of three operations: a rotation (represented by V{T), a scaling along the coordinate axes (represented by Σ), and another rotation (represented by U). Computation and Algorithms: There are several algorithms to compute the SVD of a matrix, such as the power method, QR algorithm, and Jacobi - type algorithms. These algorithms iteratively converge to the singular values and vectors of the matrix. Project Reference: https://en.wikipedia.org/wiki/Singular_value_decomposition https://www.omnicalculator.com/math/svd
