Page 370 - A Course in Linear Algebra with Applications

354 Chapter Nine: Advanced Topics

for any vector X. Let Vi denote the set of all elements of the
n
form gi(T)X with X a vector in C . Then Vj is a subspace
and the above equation for X tells us that

C n = Vi + • • • + V r.

Now in fact C n is the direct sum of the subspaces Vj, which
amounts to saying that the intersection of a Vi and the sum
of the remaining Vj, with j 7^ i, is zero. To see why this
is true, take a vector X in the intersection. Observe that
9i{T)gj{T) = 0 if % ^ j since every factor x — dk is present in
the polynomial giQj. Therefore gk(T)(X) = 0 for all k. Since
X = £ L i b kg k(T)(X), it follows that X = 0. Hence C n is
the direct sum
n
C = Vx®---®Vr.
Now the effect of T on vectors in Vi is merely to multiply
/
them by d; since (T - d^g^T) = (T) = 0. Therefore, if we
choose bases for each subspace V\,..., V r and combine them
n
to form a basis of C , then T will be represented by a diagonal
matrix. Consequently A is similar to a diagonal matrix.

Example 9.4.5
The matrix

2
has minimum polynomial (x — 2) , as we saw in Example 9.4.1.
Since this is not a product of distinct linear factors, the matrix
cannot be diagonalized.