Page 377 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 361

Another use of Jordan form is to determine which matri-
ces satisfy a given polynomial equation.

Example 9.4.8
Find up to similarity all complex n x n matrices A satisfying
the equation A 2 = I.
Let N be the Jordan normal form of A, and write N =
1 2
S^AS. Then N 2 = S~ A S. Hence A 2 = I if and only if
2
/
N = . Since N consists of a string of Jordan blocks down the
diagonal, we have only to decide which Jordan blocks J can
satisfy J 2 = i". This is easily done. Certainly the diagonal
entries of J will have to be 1 or — 1. Furthermore, matrix
multiplication reveals that J 2 ^ I if J has two or more rows.
Hence the block J must be 1 x 1. Thus N is a diagonal
matrix with all its diagonal entries equal to +1 or — 1. After
reordering the rows and columns, we get a matrix of the form

where r + s = n. Therefore A 2 = 1 if and only if A is similar
to a matrix with the form of N.
Next we consider the relationship between Jordan nor-
mal form and the minimum and characteristic polynomials.
It will emerge that knowledge of Jordan form permits us to
write down the minimum polynomial immediately. Since in
principle we know how to find the Jordan form - by using the
method of Example 9.4.7 - this leads to a systematic way of
computing minimum polynomials, something that was lacking
previously.
Let A be a complex nxn matrix whose distinct eigenval-
c
ues are i,..., c r. For each Q there are corresponding Jordan
blocks in the Jordan normal form N of A which have Q on
their principal diagonals, say Jn,..., Ju t\ let n^- be the num-
ber of rows of Ja. Of course A and N have the same minimum