An N × N matrix A is said to have an eigenvector x and corresponding eigenvalue λ if A · x = λx () Obviously any multiple of an eigenvector x will also be an eigenvector, but we won’t consider such multiples as being distinct eigenvectors. (The zero vector is not considered to be an eigenvector at all.) | Chapter 11. Eigensystems Introduction An N x N matrix A is said to have an eigenvector x and corresponding eigenvalue X if A x Ax Obviously any multiple of an eigenvector x will also be an eigenvector but we won t consider such multiples as being distinct eigenvectors. The zero vector is not considered to be an eigenvector at all. Evidently can hold only if det A - X11 0 which if expanded out is an Nth degree polynomial in A whose roots are the eigenvalues. This proves that there are always N not necessarily distinct eigenvalues. Equal eigenvalues coming from multiple roots are called degenerate. Root-searching in the characteristic equation is usually a very poor computational method for finding eigenvalues. We will learn much better ways in this chapter as well as efficient ways for finding corresponding eigenvectors. The above two equations also prove that every one of the N eigenvalues has a not necessarily distinct corresponding eigenvector If A is set to an eigenvalue then the matrix A - A1 is singular and we know that every singular matrix has at least one nonzero vector in its nullspace see on singular value decomposition . If you add tx to both sides of you will easily see that the eigenvalues of any matrix can be changed or shifted by an additive constant t by adding to the matrix that constant times the identity matrix. The eigenvectors are unchanged by this shift. Shifting as we will see is an important part of many algorithms for computing eigenvalues. We see also that there is no special significance to a zero eigenvalue. Any eigenvalue can be shifted to zero or any zero eigenvalue can be shifted away from zero. Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 456 Introduction 457 Definitions and Basic Facts A matrix is called symmetric if it is equal to its transpose A AT or aj aji It is called Hermitian or self-adjoint if it equals the complex-conjugate
