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48 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY
For the other extreme consider a single full block (∆ = {∆ | ∆ ∈C
n×n
}); the definition
of µ is now the same as that for the maximum singular value,
∆ = {∆ | ∆ ∈C
n×n
}⇒µ(M)=σ
max
(M).
Observe that every possible block structure, ∆, contains {λI | λ ∈C}as a perturbation;
and every possible block structure, ∆, is contained in C
n×n
. These particular block
structures are the boundary cases. This means that the resulting µ tests act as bounds
on µ for any block structure, ∆. This gives the following bounds.
ρ(M) ≤ µ(M) ≤ σ
max
(M).
The above bounds can be arbitrarily conservative but can be improved by using the
following transformations. Define the set
D =
diag(D
1
,...,D
q
,d
1
I
1
,...,d
m
I
m
,)
D
j
=D
∗
j
>0,
dim(I
i
)=k
i
,d
i
∈R,d
i
>0
.(2.15)
This is actually the set of invertible matrices that commute with all ∆ ∈ ∆. This allows
us to say that for all D ∈Dand for all ∆ ∈ ∆,
D
−1
∆D =∆.
Packard [3] shows that the restriction that d
i
be positive real is without loss of
generality. We can actually take one of these blocks to be one (or the identity).
Now define Q as the set of unitary matrices contained in ∆:
Q =
Q ∈ ∆
Q
∗
Q = I
. (2.16)
The sets D and Q can be used to tighten the bounds on µ in the following way (refer to
Doyle [68]).
max
Q∈Q
ρ(QM) ≤ µ(M) ≤ inf
D∈D
σ
max
(DMD
−1
). (2.17)