Risk Management and Financial Institutions, + Web Site, 3rd Edition

APPENDIX H

Eigenvectors and Eigenvalues

Consider an n × n matrix A and suppose that x is an n × 1 vector. Consider the equation

(H.1)

The equation can be written

where I is the n × n identity matrix (which is the n × n matrix with diagonal elements equal to 1 and all other elements equal to zero). Clearly, x=0 is a solution to equation (H.1). Under what circumstances are there other solutions? A theorem in linear algebra tells us that there are other solutions when the determinant of A - λ I is zero. The values of λ that lead to solutions of equation (H.1) are therefore the values of λ that we get when we solve the equation that sets the determinant of A - λ I equal to zero. This equation is a nth order polynomial in λ. In general, it has n solutions. The solutions are the eigenvalues of the matrix, A. The vector x that solves equation (H.1) for a particular eigenvalue is an eigenvector. In general, there are n eigenvectors, one corresponding to each eigenvalue.

As a simple example, suppose that

In this case

The determinant of this matrix is

The solutions to this equation are λ = ...