Iterative Optimization in Inverse Problems

138 Iterative Optimization in Inverse Problems

Corollary 9.7 If 0 <γ

i

≤ p

−1

i

for all i, then the matrix B with entries

B

ij

=

√

γ

i

A

ij

has ρ(B

†

B) ≤ 1.

Proof: We have

J



j=1

s

j

|B

ij

|

2

= γ

i

J



j=1

s

j

|A

ij

|

2

= γ

i

p

i

≤ 1.

Therefore, ρ(B

†

B) ≤ 1, according to the theorem.

Corollary 9.8 If, for some a in the interval [0, 2], we have

α

i

≤ r

−1

ai

, (9.36)

for each i,and

β

j

≤ c

−1

aj

, (9.37)

for each j, then, for the matrix G with entries

G

ij

= A

ij

√

α

i



β

j

,

no eigenvalue of G

†

G exceeds one.

Proof: We calculate c

aj

(G)andr

ai

(G) and find that

c

aj

(G) ≤



max

i

α

a/2

i



β

a/2

j

I



i=1

|A

ij

|

a

=



max

i

α

a/2

i



β

a/2

j

c

aj

(A),

and

r

ai

(G) ≤



max

j

β

1−a/2

j



α

1−a/2

i

r

ai

(A).

Therefore, applying the inequalities (9.36) and (9.37), w