V
n
¼ W A È W A :
Notice that if we consider an identity operator E as an idempotent operator, we are
thinking of a trivial case. That is, V
n
¼ V
n
È {0}.
The result can immediately be extended to the case where more idempotent
operators take part in the vector space. Let us define operators such that
A 1 þ A 2 þ Á Á Á þ A s ¼ E,
where E is a (n, n) identity matrix. Also
A i A j ¼ A i δ ij :
Moreover, we define W i such that
W i ¼ A i V
n
¼ x; x 2 V
n , A i x ¼ x
f
g :
ð12:77Þ
Then
V
n
¼ W 1 È W 2 È Á Á Á È W s :
ð12:78Þ
In fact, suppose that
∃ x( 2 V
n ) 2 W i , W j (i 6 ¼ j). Then A i x ¼ x = A j x. Operating A j
( j 6 ¼ i) from the left, A j A i x ¼ A j x = A j A j x. That is, 0 ¼ x = A j x, implying that
W i \ W j ¼ {0}. Meanwhile,
V
n
¼ A 1 þ A 2 þ Á Á Á þ A s
ð
Þ V
n
¼ A 1 V
n
þ A 2 V
n
þ Á Á Á þ A s V
n
¼ W 1 þ W 2 þ Á Á Á þ W s :
ð12:79Þ
As W i \ W j ¼ {0} (i 6 ¼ j), (12.79) is a direct sum. Thus, (12.78) will follow.
Example 12.5 Think of the following transformation A:
e 1 e 2 e 3 e 4
ð
Þ A ¼ e 1 e 2 e 3 e 4
ð
Þ
1 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
0
B
B
B
@
1
C
C
C
A
¼ e 1 e 2 0 0
ð
Þ :
Put x =
P 4
i¼1 x i e i . Then, we have
A x
ð Þ ¼
X 4
i¼1
x i A e i
ð Þ =
X 2
i¼1
x i e i = W A ,
E À A
ð
Þ x
ð Þ ¼ x 2 A x
ð Þ =
X 4
i¼3
x i e i = W A :
In the above,
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12 Canonical Forms of Matrices