CHAPTER 5 - Decomposition of Matrix Transformations: Eigenstructures and Quadratic Forms

Abstract

This chapter discusses decomposition of matrix transformations and describes eigenstructures and quadratic forms. Vectors, which under a given transformation map into themselves or multiples of themselves, are called invariant vectors under that transformation. The distinguishing feature of an eigenvector basis is that the nature of the transformation assumes a particularly simple geometric form. If a symmetric matrix consists of all real-valued entries, then all of its eigenvalues and associated eigenvectors can be real valued. Eigenstructures are computed only for square matrices. The rank of either a minor or major product moment is the same as the rank of the matrix itself. Finding the eigenstructure of a matrix—either symmetric to begin with or else a derived product-moment matrix—is a highly useful procedure for determining matrix rank. Data-based product-moment matrices exhibit a number of virtues, such as real-valued eigenstructures and orthogonal transformations for rotating the matrix to diagonal form.