ECE 598GG Fall 2007
Prof. George Gross
Room 339 Everitt Lab
Homework 8
due Tuesday, October 30, 2007
1. [40 points] We have a set of m observations z. It is known that
z = H x(*)
where, H
∈
m x n
and x
∈
n
with m > n. We wish to determine the “best” solution
using a weighted-least-squares or WLS criterion. Derive
ˆ
x
, the WLS solution of (*), using
the criterion of minimizing the weighted sum of the squares of residuals
1
2 [z – H x ]T R-1 [z – H x ]
Here R-1
∈
m x m
is a diagonal matrix with positive entries. State any and all assumptions
that you require for your solution to work.
2. [30 points] An additional observation z a
= hT x , where, h
∈
n
, has just been made
available. This new information is added to the existing observations with a weight of r-2.
Show that the WLS solution
x
∧
a
of the augmented system
hT
H
⎡
⎣
⎢⎤
⎦
⎥ x = za
z
⎡
⎣
⎢⎤
⎦
⎥
(**)
using the weighting matrix
r−2
0
0
R−1
⎡
⎣
⎢⎤
⎦
⎥
is
x
∧
a
= x
∧
+
za−hTx
∧
⎡
⎣
⎢⎤
⎦
⎥
r2+hT
Σ
h
Σ
h
where
Σ
Δ
(HT R-1 H)-1
.
Hint: The Sherman-Morrison-Woodbury formula is a convenient expression for the
inverse of ( A + U V T ). Let A
∈
n x n
be nonsingular and let U, V
∈
n x k
.
Assume [ I + VT A-1
U ] is nonsingular. Then,
[ A + U V T ]-1
= A-1 – A-1 U [ I + V T A-1 U ]-1
V T A-1
3. [30 points] Prove the results of the Example on p.1 of the Least-Squares Solution Notes.