Diagonalization

Abstract

Gaussian elimination plays a fundamental role in solving a system Ax = b of linear equations. In general, instead of solving the given system, one could try to solve the normal equation A^T Ax = A^Tb, whose solutions are the true solutions or the least squares solutions depending on whether or not the given system is consistent. Note that the matrix A^T A is a symmetric square matrix, and so one may assume that the matrix in the system is a square matrix. For this kind of reason, we focus on a diagonal matrix or a linear transformation from a vector space to itself throughout this chapter.