With just a little bit more work we shall arrive at a similar
expansion for BB itself. We begin
by applying the second resolvent identity to
Pj
Pj
. More precisely, we note that the second resolvent
identity implies that
B
Pj
=
Pj
B=∫
C
j
zRz−Id
z
B
Pj
Pj
B
z
C
j
z
R
z
I
(1)
Pj
B=∫
C
j
zRzd
z
Pj
B
z
C
j
z
R
z
Pj
B=∫
C
j
Rz(z−
λ
j
)d
z
+
λj
∫
C
j
Rzd
z
Pj
B
z
C
j
R
z
z
λ
j
λj
z
C
j
R
z
Pj
B=
Dj
+
λj
Pj
Pj
B
Dj
λj
Pj
Summing this over
jj we find
B∑
j
=1h
Pj
=∑
j
=1h
λj
Pj
+∑
j
=1h
Dj
B
j
1
h
Pj
j
1
h
λj
Pj
j
1
h
Dj
(2)
We can go one step further, namely the evaluation of the first
sum. This stems from
the eqn in the discussion of the transfer
function where we integrated
Rs
Rs
over a circle
Cρ
Cρ
where
ρ>∥B∥
ρ
norm
B
. The connection to the
Pj
Pj
is made by the residue theorem. More precisely,
∫
Cρ
Rzd
z
=2πi∑
j
=1h
Pj
z
Cρ
R
z
2
j
1
h
Pj
Comparing this to
the eqn from the discussion of the transfer
function we find
∑
j
=1h
Pj
=I
j
1
h
Pj
I
(3)
and so
Equation 2 takes the form
B=∑
j
=1h
λj
Pj
+∑
j
=1h
Dj
B
j
1
h
λj
Pj
j
1
h
Dj
(4)
It is this formula that we refer to as the
Spectral
Representation of
BB. To the
numerous connections between the
Pj
Pj
and
Dj
Dj
we wish to add one more. We first write
Equation 1 as
(B−
λj
I)
Pj
=
Dj
B
λj
I
Pj
Dj
and then raise each side to the
mj
mj
power. As
P
j
m
j
=
Pj
P
j
m
j
Pj
and
D
j
m
j
=0
D
j
m
j
0
we find
B−
λj
I
m
j
Pj
=0
B
λj
I
m
j
Pj
0
(5)
For this reason we call the range of
Pj
Pj
the
jjth
generalized
eigenspace, call each of its nonzero members a
jjth
generalized
eigenvector and refer to the dimension of
ℛ
Pj
ℛ
Pj
as the
algebraic multiplicity of
λj
λj
. With regard to
the first
example from the discussion of the eigenvalue
problem, we note that although it has only two linearly
independent eigenvectors the span of the associated
generalized eigenspaces indeed fills out
R3
3
. One may view this as a consequence of
P1
+
P2
=I
P1
P2
I
, or, perhaps more concretely, as appending the generalized
first eigenvector
(
010
)T
010
to the original two eigenvectors
(
100
)T
100
and
(
001
)T
001
. In still other words, the algebraic multiplicities
sum to the ambient dimension (here 3), while the sum of
geometric multiplicities falls short (here 2).
