Mathematics for Physicists

10 Eigenvalues and eigenvectors

Given a square matrix A, it is often required to find scalar constants λ and vectors x such that

(10.1)

is satisfied. This equation only has non-trivial solutions x ≠ 0 for particular values of λ. These values are called eigenvalues and the corresponding vectors x are called eigenvectors.¹ In physical applications the eigenvalues often correspond to the allowed values of observable quantities. In what follows, we shall firstly consider the solutions of (10.1) in general, before specialising to Hermitian matrices, which are the most important in physical applications. We then show how knowledge of the eigenvalues can be used to transform the matrix A to diagonal form, with applications to the theory of small vibrations and geometry.

10.1 The eigenvalue equation

The eigenvalue equation (10.1) may be written in the form

(10.2)

This is a set of homogeneous linear simultaneous equations in the components x_i (i = 1, 2, …, n) of the type discussed in Section 9.1.2 and has non-trivial solutions if, and only if,

(10.3)

which is called the characteristic equation of the matrix A. The determinant is given by

(10.4a)

where

(10.4b)

is a polynomial ...