Page 378 - A Course in Linear Algebra with Applications

362 Chapter Nine: Advanced Topics

and characteristic polynomials since they are similar matrices.
Now J^ is an n^ x n^ upper triangular matrix with ci on
the principal diagonal, so its characteristic polynomial is just
ni
(ci — x) i. The characteristic polynomial p of N is clearly the
product of all of these polynomials: thus

r h
n
p — TT(CJ — x) mi where rrii = \ J i j -
i = i j=i

The minimum polynomial is a little harder to find. If /
is any polynomial, it is readily seen that f(N) is the matrix
with the blocks f(Jij) down the principal diagonal and zeros
/
elsewhere. Thus f(N) = 0 if and only if all the ( J ^ ) = 0.
Hence the minimum polynomial of N is the least common
multiple of the minimum polynomials of the blocks J^. But
we saw in Example 9.4.6 that the minimum polynomial of the
nij
Jordan block J^ is (x — Ci) . It follows that the minimum
polynomial of iV is

n c fci
/=no* - *)

where ki is the largest of the n^ for j = 1,..., U.
These conclusions, which amount to a method of com-
puting minimum polynomials from Jordan normal form, are
summarized in the next result.
Theorem 9.4.8
Let A be an nxn complex matrix and let c±,... ,c r be the dis-
tinct eigenvalues of A. Then the characteristic and minimum
polynomials of A are
n n
Y[(ci-x) mi and Y[( - ) ki
x Ci
i = l i=l