Page 368 - A Course in Linear Algebra with Applications

352 Chapter Nine: Advanced Topics

characteristic polynomial. Therefore it is sufficient to prove
the statement for the triangular matrix T. From Example 8.1.2
we know that the characteristic polynomial of T is

(in -x)---(t nn -x).
On the other hand, direct matrix multiplication shows that
(tnl — T) • • • (t nnI — T)=0: the reader may find it helpful to
check this statement for n — 2 and 3. The result now follows
from 9.4.1.
At this juncture the reader may wonder if the minimum
polynomial is really of much interest, given that it is a divisor
of the more easily calculated characteristic polynomial. But in
fact there are features of a matrix that are easily recognized
from its minimum polynomial, but which are unobtainable
from the characteristic polynomial. One such feature is diag-
onalizability.

Example 9.4.4
(\ \ \
Consider for example the matrices I2 and I I: both of

2
these have characteristic polynomial (x — ) , but the first
l
matrix is diagonalizable while the second is not. Thus the
characteristic polynomial alone cannot tell us if a matrix is
diagonalizable. On the other hand, the two matrices just con-
2
sidered have different minimum polynomials, x — 1 and (x — )
l
respectively.
This example raises the possibility that it is the mini-
mum polynomial which determines if a matrix is diagonaliz-
able. The next theorem confirms this.
Theorem 9.4.4
Let A be an n x n matrix over C. Then A is diagonalizable if
and only if its minimum polynomial splits into a product of n
distinct linear factors.