Page 282 - A Course in Linear Algebra with Applications

266 Chapter Eight: Eigenvectors and Eigenvalues

On the other hand, one cannot expect similar matrices to
have the same eigenvectors. Indeed the condition for X to be
an eigenvector of SAS~ X with eigenvalue c is (SAS'^X —
1
cX, which is equivalent to Atf^X) = c(S~ X). Thus X is
X
an eigenvector of SAS~~ l if and only if S~ X is an eigenvector
of A

Eigenvectors and eigenvalues of linear transformations

Because of the close relationship between square matri-
ces and linear operators on finite-dimensional vector spaces
observed in Chapter Six, it is not surprising that one can also
define eigenvectors and eigenvalues for a linear operator.
Let T : V —» V be a linear operator on a vector space
V over a field of scalars F. An eigenvector of T is a non-zero
vector v of V such that T(v) = cv for some scalar c in F:
here c is the eigenvalue of T associated with the eigenvector
v.
Suppose now that V is a finite-dimensional vector space
over F with dimension n. Choose an ordered basis for V, say
B. Then with respect to this ordered basis T is represented
by an n x n matrix over F, say A; this means that

= A[v] B.
[T(y)] B

Here [U]B is the coordinate column vector of a vector u in V
with respect to basis B . The condition T(v) = cv for v to
be an eigenvector of T with associated eigenvalue c, becomes
-AMB = c[v]g, which is just the condition for M s to be an
eigenvector of the representing matrix A; also the eigenvalues
of T and A are the same.
If the ordered basis of V is changed, the effect is to replace
A by a similar matrix. Of course any such matrix will have
the same eigenvalues as T; thus we have another proof of the
fact that similar matrices have the same eigenvalues.