Matrices: a brief overview - Signal Processing and Optimization for Transceiver Systems - page 759

B.4.2 Matrix inversion lemma

Let P and R be square and nonsingular. Then, according to the matrix inversion
lemma, we have
( P + QRS ) − 1 = P − 1 − P − 1 Q ( SP − 1 Q + R − 1 ) − 1 SP − 1 .

(B . 13)

This holds even if Q and S are not square. To prove Eq. (B.13) just observe
that

P − 1 − P − 1 Q ( SP − 1 Q + R − 1 ) − 1 SP − 1 ( P + QRS )

I − P − 1 Q ( SP − 1 Q + R − 1 ) − 1 S + P − 1 Q ( SP − 1 Q + R − 1 ) − 1 SP − 1 Q = I − RS

I − P − 1 Q ( SP − 1 Q + R − 1 ) − 1 S
+ P − 1 Q ( SP − 1 Q + R − 1 ) − 1

=

SP − 1 Q + R − 1 − SP − 1 Q

RS

I − P − 1 Q ( SP − 1 Q + R − 1 ) − 1 S + P − 1 Q ( SP − 1 Q + R − 1 ) − 1 S = I

=

indeed.

B.4.3 Partitioned matrices

Consider a matrix in the form

AB
CD

.

(B . 14)

Assuming that A is nonsingular, we can verify by explicit multiplication that
AB
CD

= A 0

0D − CA − 1 B

IA − 1 B
0I

.

I 0

(B . 15)

CA − 1 I

By recognizing that

− 1 = IA − 1 B

0I

− 1 = I − A − 1 B
0

I 0 I 0

and

CA − 1 − CA − 1 I I I

we therefore obtain
AB
CD

− 1 = I − A − 1 B
0

A − 1 ,

0 I 0

− CA − 1

( D − CA − 1 B ) − 1

I 0 I

(B . 16)
where it is assumed that D − CA − 1 B is nonsingular as well. Similarly, for the
case where D is nonsingular, we can write
AB
CD

= IBD − 1
0I

A − BD − 1 C0
0

I 0
D − 1 CI

,

(B . 17)

D

which yields
AB
CD

− 1 = I 0
− D − 1 CI

( A − BD − 1 C ) − 1 I − BD − 1

0

.

0

D − 1 0 I

(B . 18)