ECE 497-3: Inverse Problems in Engineering
Homework #2
Due: Friday December 20, 2002
Motivation
In this homework, we will look at the singular value decomposition of a matrix as a method
for examining least squares problems. In this homework, the attatched Matlab routines will
do most of the work. You will need to set the “numeric format” (in the options window) to
“long” for this homework.
1) Consider the following linear system of equations
y=
−0.2042 0.0559 0.3715 −0.2077
0.1347 0.2208 0.5294 0.1336
0.4803 0.3889 0.6862 0.4670
0.6078 0.6088 1.2159 0.6081
x
Using Matlab, compute yfor xT= [1 1 1 1]. Then, using this y, compute xby determining
the inverse of the above matrix.
It turns out that for any matrix Z∈Rm×nwe can always compute its singular value decom-
position
Z=USVT
where, U∈Rm×mand Uis a unitary matrix (UT=U−1), V∈Rn×nand Vis a unitary
matrix (VT=V−1), and S∈Rm×n.Sis all zeros except for a submatrix, which will will
denote Σ, of size m×mor n×n(depending on whether mor nis smaller) in the upper left
corner. The entries in the submatrix Σ are called the singular values of the matrix and will
b e denoted σifor the ith singlular value. In general, σi≥σi+1 , that is, the singular values
decrease as the mode number increases (this is important!) As you will see, these singular
values contain a great deal of information about the matrix (and what may be going wrong).
2) Lets assume that
x=Vα
y=Uβ
That is, xcan be written as a linear combination of the columns of V
x=Vα
= [v1v2... vn]α
=v1α1+v2α2+... +vnαn
Since x∈Rnand the columns of Vspan Rn, we know such an expansion must exist. Simi-
larly, we must be able to represent yas a linear combination of the columns of U.
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