Page 369 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 353

Proof
1
Assume first that A is diagonalizable, so that S~ AS — D, a
diagonal matrix, for some invertible S. Then A and D have
the same minimum polynomials by 9.4.2. Let di,...,d r be
the distinct diagonal entries of D; then Example 9.4.2 shows
that the minimum polynomial of D is (x — d\) • • • (x — d r),
which is a product of distinct linear factors.
Conversely, suppose that A has minimum polynomial

/ = (x-di)---(x-d r)

where di,... ,d r are distinct complex numbers. Define #; to be
the polynomial obtained from / by deleting the factor x — di.
Thus
/
9i = -}-•
x — di
Next we recall the method of partial fractions, which is
useful in calculus for integrating rational functions. This tells
us that there are constants b\,..., b r such that

I = V^ b i
f ~ ^ x- d{

Multiplying both sides of this equation by , we obtain
/

1 = M i -\ r- Kg r

by definition of gi.
At this point we prefer to work with linear operators, so
we introduce the linear operator T on C n defined by T(X) —
AX. It follows from the above equation that b\g\{T) + • • • +
b rg r(T) is the identity function. Hence

X = b l9l(T)(X) + --- + b rg r(T)(X)