The Cooper Union Department of Electrical Engineering
ECE416 Adaptive Filters
Problem Set I: Linear Algebra
Note: Unless stated otherwise, consider only Önite dimensional linear spaces. Also assume
complex values.
Aside from basic MATLAB functions, in this homework you are allowed to use: rank, orth,
svd, svds, eig, eigs, pinv, chol. Use doc to familiarize yourself with these functions. Given
that these functions are available to you, I want you to use e¢ cient code. Do NOT Önd more
advanced or more sophisticated functions.
Notation: For a matrix A, RA represents the range space and NA the null-space. If K is a
linear space, PK is the orthogonal projection onto that space.
Matrix inversion lemma:
B
1 + CD1C
H
1
= B BC
D + C
HBC1
C
HB
1. Let A be a square invertible matrix, a scalar and u a vector. Use the matrix inversion
lemma to simplify:
A + u uH
1
For certain values of , this matrix may not be invertible; assume this is not the case.
Now take the special case where A = I. The answer reduces to something of the form
c1I + c2u uH. Find this expression. Also, for this case, determine the condition on
for this to be invertible.
2. Let fuig
r
i=1 be r linearly independent M 1 vectors, and fvig
r
i=1 be r linearly independent N 1 vectors. Show that the M N matrix A given by:
A =
Xr
i=1
uiv
H
i
has rank exactly r. Hint: Think about RA and N ?
A .
3. Let ~x; ~u be M 1 column vectors. We want to Önd optimal such that ~x ~u.
(a) Find u
#. Use your formula to show that:
=
hx; ui
juj
2
(b) Letís generalize this. First: if D = diag fdig, does DA or AD scale the columns
of A by diís? So then what scales the rows?
(c) Let U be an M N matrix with orthogonal (not necessarily orthonormal) columns
fuig. Find the SVD for U, from that Önd U
#, and use this to solve the following
problem: given x, Önd coe¢ cients f ig such that:
x
X iui
Note: You should know the answer. Here we are using the pseudo-inverse to
conÖ
