Page 283 - A Course in Linear Algebra with Applications

8.1: Basic Theory of Eigenvectors 267

These observations permit us to carry over to linear op-
erators concepts such as characteristic polynomial and trace,
which were introduced for matrices.

Example 8.1.4
Consider the linear transformation T : Doo[a,b] —> Doo[a,6]
/
where T(f) = ', the derivative of the function /. The con-
/
dition for / to be an eigenvector of T is ' = cf for some
constant c. The general solution of this simple differential
equation is / = de cx where d is a constant. Thus the eigen-
values of T are all real numbers c, while the eigenvectors are
the exponential functions de cx with d ^ 0.

Diagonalizable matrices
We wish now to consider the question: when is a square
matrix similar to a diagonal matrix? In the first place, why
is this an interesting question? The essential reason is that
diagonal matrices behave so much more simply than arbitrary
matrices. For example, when a diagonal matrix is raised to
the nth power, the effect is merely to raise each element on
the diagonal to the nth power, whereas there is no simple
expression for the nth power of an arbitrary matrix. Suppose
that we want to compute A n where A is similar to a diagonal
X
matrix D, with say A = SDS~ . It is easily seen that A n =
1
n
SD S~ . Thus it is possible to calculate A n quite simply
if we have explicit knowledge of S and D. It will emerge in
8.2 and 8.3 that this provides the basis for effective methods
of solving systems of linear recurrences and linear differential
equations.
Now for the important definition. Let A be a square
matrix over a field F. Then A is said to be diagonalizable over
F if it is similar to a diagonal matrix D over F, that is, there
is an invertible matrix S over F such that A = SDS -1 or
1
equivalently, D = S~ AS. One also says that S diagonalizes
A. A diagonalizable matrix need not be diagonal: the reader