Page 273 - A Course in Linear Algebra with Applications
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Chapter Eight
EIGENVECTORS AND EIGENVALUES
An eigenvector of an n x n matrix A is a non-zero n-
column vector X such that AX = cX for some scalar c, which
is called an eigenvalue of A. Thus the effect of left multiplica-
tion of an eigenvector by A is merely to multiply it by a scalar,
and when n < 3, a parallel vector is obtained. Similarly, if T
is a linear operator on a vector space V, an eigenvector of T is
a non-zero vector v of V such that T(v) = cv for some scalar
3
c called an eigenvalue. For example, if T is a rotation in R ,
the eigenvectors of T are the non-zero vectors parallel to the
axis of rotation and the eigenvalues are all equal to 1.
A large amount of information about a matrix or linear
operator is carried by its eigenvectors and eigenvalues. In
addition, the theory of eigenvectors and eigenvalues has im-
portant applications to systems of linear recurrence relations,
Markov processes and systems of linear differential equations.
We shall describe the basic theory in the first section and
then we give applications in the following two sections of the
chapter.
8.1 Basic Theory of Eigenvectors and Eigenvalues
We begin with the fundamental definition. Let A be an
n x n matrix over a field of scalars F. An eigenvector of A is
a non-zero n-column vector X over F such that
AX = cX
for some scalar c in F; the scalar c is then referred to as the
eigenvalue of A associated with the eigenvector X.
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