Project 4 A Recovered 1

2010

2002

Matrix theory and its applications make wide use of the eigenprojections of square matrices. The present paper demonstrates that the eigenprojection of a matrix A can be calculated with the use of any annihilating polynomial of A u , where u ≥ ind A. This enables one to find the components and the minimal polynomial of A, as well as the Drazin inverse A D .

W e will cover the basics of matrix algebra here. We will (i) describe what a matrix is, and then discuss (ii) matrix addition, (iii) matrix multiplication by a constant, (iv) multiplication of matrices, (v) the identity matrix, (vi) the inverse of a matrix, (vii) an algorithm for finding the inverse of a matrix, and (viii) the transpose of a matrix. We will then use matrix algebra to (ix) solve the simple problem of fitting a straight line between two points, and (x) the slightly more complex problem of fitting a straight line to three points. Finally, we will discuss (xi) eigenvectors and eigenvalues, including (xii) an algorithm to find them. While on the topic of matrix algebra we will describe three Excel commands you should probably know if you are going to manipulate matrices, and show a nice trick for exporting data from Excel into MATLAB. Excel and MATLAB are powerful tools for manipulating numerical data. To test your understanding of matrix algebra, a few problems (with solutions) are then presented.

Applied Linear Algebra in Action, 2016

In natural sciences and engineering, are often used differential equations and systems of differential equations. Their solution leads to the problem of eigenvalues. Because of that, problem of eigenvalues occupies an important place in linear algebra. In this caption we will consider the problem of eigenvalues, and to linear and quadratic problems of eigenvalues. During the studying of linear problem of eigenvalues, we put emphasis on QR algorithm for unsymmetrical case and on minmax characterization of symmetric case. During the studying of quadratic problems of eingenvalue, we consider the linearization and variational characterization. We illustrate all with practical examples.

Preface ix • To reiterate, we have added approximately 20% more exercises, most elementary and computational in nature. We have included more solved problems at the back of the book and, in many cases, have added similar new exercises. We have added some additional blue boxes, as well as a table giving the locations of them all. And we have added more examples early in the text, including more sample proof arguments.