380 CHAPTER 13. MATRICES AND THE INNER PRODUCT
must be Hermitian. Therefore, it is necessary that(σ 00 0
)(P QR S
)=
(σP σQ0 0
)=
(I σQ0 0
)is Hermitian. Then (
I σQ0 0
)=
(I 0
Q∗σ 0
)and so Q = 0.
Next,
A0︷ ︸︸ ︷V
(P QR S
)U∗
A︷ ︸︸ ︷UΣV ∗ =V
(Pσ 0Rσ 0
)V ∗ =V
(I 0
Rσ 0
)V ∗
is Hermitian. Therefore, also (I 0
Rσ 0
)is Hermitian. Thus R = 0 because(
I 0Rσ 0
)∗=
(I σ∗R∗
0 0
)
which requires Rσ = 0. Now multiply on right by σ−1 to find that R = 0.Use 13.37 and the second equation of 13.36 to write
A0︷ ︸︸ ︷V
(P QR S
)U∗
A︷ ︸︸ ︷UΣV ∗
A0︷ ︸︸ ︷V
(P QR S
)U∗ =
A0︷ ︸︸ ︷V
(P QR S
)U∗.
which implies (P QR S
)(σ 00 0
)(P QR S
)=
(P QR S
).
This yields from the above in which is was shown that R,Q are both 0(σ−1 0
0 S
)(σ 00 0
)(σ−1 0
0 S
)=
(σ−1 0
0 0
)(13.39)
=
(σ−1 0
0 S
). (13.40)
Therefore, S = 0 also and so
V ∗A0U ≡
(P QR S
)=
(σ−1 0
0 0
)