Nature is rich in symmetry and many of the matrices which arise from physical problems will have some kind of symmetry features. Usually these matrices will be real symmetric. The theory of real symmetric matrices can be regarded as a special case of the theory of complex hermitian matrices. We will examine various aspects and applications of the theory of real symmetric and complex hermitian matrices.

We begin the chapter by discussing Schur’s unitary triangularization theorem for complex matrices. We introduce the ideas of quadratic and hermitian forms and the way in which real symmetric and complex hermitian matrices arise from these forms. Topics which we treat include the notionof positive-definiteness for forms and matrices, the definition and calculation of the signature of a form, and the simultaneous reduction of a pair of forms. We go on to consider the eigenvalues of symmetric and hermitian matrices. The symmetry of these matrices makes available techniques different from those encountered earlier in this book In particular we consider the Rayleigh quotient from which estimates for the largest and smallest eigenvalues of a symmetric or hermitian matrix can be obtained, Rayleigh’s principle which is used for estimating other eigenvalues, the Courant-Fisher min-max theorem and various applications.

Matrix Theory DAVID W. LEWIS Department of Mathematics University College Dublin Ireland u>World Scientific Singapore • New Jersey * London • Hong kong. Program Za Spajanje Dvije Slike U Jednu here. This book provides an introduction to matrix theory and aims to provide a clear and concise. Download Pdf. Share this article. Matrix Theory by David W.

This book provides an introduction to matrix theory and aims to provide a clear and concise exposition of the basic ideas, results and techniques in the subject. Complete proofs are given, and no knowledge beyond high school mathematics is necessary. The book includes many examples, applications and exercises for the reader, so that it can used both by students interested in theory and those who are mainly interested in learning the techniques. Sample Chapter(s) Contents: • Matrices and Linear Equations • Vector Spaces and Linear Maps • Matrix Norms • Eigenvalues and Eigenvectors • Jordan Canonical Form and Applications • Symmetric and Hermitian Matrices • Perturbation Theory • Further Topics Readership: Students in mathematics, engineering and science.