Matrix Decomposition

Abstract

Matrices are divided into different classes, depending on the form and specific properties of the matrix. Matrix factorizations depend on the properties of certain class of matrices, hence matrix factorization are of great importance in the matrix theory, in the analysis of numerical algorithms and even in numerical linear algebra. A factorization of the matrix A is a representation of A as a product of several "simpler" matrices, which makes the problem at hand easier to solve. Factorizations of matrices into some special sorts of matrices with similarity are of fundamental importance in matrix theory, like Schur decomposition, spectral decomposition and the singular value decomposition. Furthermore, the basic tool for solving systems of linear equations, as one of the basic problems of numerical linear algebra, is the LU factorization. Also, it is important to mention QR factorization and its calculation through rotation and reflectors.