Page 367 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 351

In the computation of minimum polynomials the next
result is very useful.
Lemma 9.4.2
Similar matrices have the same minimum polynomial.

The quickest way to see this is to recall that similar ma-
trices represent the same linear operator, and hence their min-
imum polynomials equal the minimum polynomial of the lin-
ear operator. Thus, by combining Lemma 9.4.2 and Example
9.4.2, we can find the minimum polynomial of any diagonal-
izable complex matrix.
Example 9.4.3
1 2
Find the minimum polynomial of the the matrix
2 1

By Example 9.1.1 the matrix is similar to
Hence the minimum polynomial of the given matrix is
(x-3)(x + l).

In Chapter Eight we encountered another polynomial as-
sociated with a matrix or linear operator, namely the charac-
teristic polynomial. It is natural to ask if there is a connection
between these two polynomials. The answer is provided by a
famous theorem.

Theorem 9.4.3 (The Cayley-Hamilton Theorem)
Let A be annxn matrix over C. Ifp is the characteristic poly-
nomial of A, then p(A) = 0. Hence the minimum polynomial
of A divides the characteristic polynomial of A.

Proof
According to 8.1.8, the matrix A is similar to an upper tri-
X
angular matrix T; thus we have S~ AS = T with S invert-
ible. By 9.4.2 the matrices A and T have the same minimum
polynomial, and we know from 8.1.4 that they have the same