19 The matrix eigenvalue problem
19.1 Concepts
The determination of the eigenvalues and eigenvectors of a square matrix is called the
eigenvalue problem, and is of importance in many branches of the physical sciences
and engineering.
1For example in quantum chemistry, the application of the variation
principle to the Schrödinger equation results in the replacement of the differential
eigenvalue equation by an equivalent matrix eigenvalue equation that can be solved
by numerical methods. We have already met the eigenvalue problem in Section 9.4,
on the use of Lagrangian multipliers for finding the stationary values of a quadratic
form (Example 9.9), and in Section 17.4, on the use of determinants for the solution
of systems of secular equations. In the present chapter, the matrix eigenvalue problem
and the properties of eigenvalues and eigenvectors are discussed in Sections 19.2 and
19.3. The closely related problems of matrix diagonalization and of the reduction of
quadratic forms to canonical form are discussed in Sections 19.4 and 19.5. Section 19.6
contains a summary of the complex matrices that are important in advanced group
theory and in quantum mechanics.
We first summarize the matrix formulation of the solution of systems of inhomo-
geneous linear equations, already discussed in Section 17.4 in terms of determinants.
A system of nlinear equations in nunknowns,x
1,x
2,x
3,=,x
n(equation (17.26)
(19.1)
can be written as the single matrix equation
(19.2)
aaa a
aaa a
aaa a
nnn11
12 13121 22 23231 32 333
aaaa a
n
nn1 nn
23=
x
x
x
x
b
n1231123b
b
b
nax
ax
ax
ax
ax
ax
ax
n11 121 131 11112 222 232 2ax
ax
ax
ax
ax
a
nn 2213 323 333 3331++
++
++
++
nnnnnnnnn n nx
ax
ax
ax
b
b
b
b
=
=
=
=
212331The earliest eigenvalue problems were considered by d’Alembert between 1743 and 1758 in connection with
the solution of systems of linear equations with constant coefficients arising from the motions of a loaded string.
Cauchy discussed eigenvalues in connection with the forms that describe quadratic surfaces. In 1829 he showed that a
quadratic form can be reduced to diagonal form (Section 19.5) by the constrained optimization method involving
Lagrangian multipliers described in Example 9.9. The multipliers are the eigenvalues of the associated matrix.