EECS 453/551: HW1 SOLUTIONS
Problem 1 (*)
m
Let e R be a vector with ei = 1 for i = 1, . . . , m. Let x Rn be a vector with xj = j for j = 1, . . . , n.
Then the desired n m matrix whose j-th row equals j is given by the outer-product xeT . In MATLAB we
w
EECS 551: HW 3
Reading for next week: Chapter 2 and Chapter 3 of Laub
Question 1. This what is known about the dietary habits of the mythical Michigan Wolverine who eats
only grapes, cheese or lettuce. The creatures dietary habits conform to the following
EECS 551 - HOMEWORK 1
Reading pertaining to the problem set: Chapter 1 of Laub
Reading for next week: Chapter 2 Section 9.1 of Laub
Problem 1. Express the n m matrix A whose j th row equals j as an outer-product of two appropriately
dened vectors.
Problem
EECS 551- HW 2
Reading pertaining to problem set: Chapter 2, Chapter 5.1, Chapter 9.1, Chapter 13.1-13.2
Reading for next week: Chapter 3, Chapter 5.2
Problem 1. How are eigenvalues and eigenvectors of B = A 10I related to the eigenvalues and
eigenvectors
EECS551: HW1 SOLUTIONS
Problem 1
m
Let e R be a vector with ei = 1 for i = 1, . . . , m. Let x Rn be a vector with xj = j for j = 1, . . . , n.
Then the desired n m matrix whose j-th row equals j is given by the outer-product xeT . In MATLAB we
would type
EECS551: HW2 SOLUTIONS
Problem 1
T
Let A = QQ be the eigendecomposition of A. Then we may write
B = A 10I
= QQT 10QQT
(I = QQT since Q is orthogonal)
= Q( 10I)QT
(0.1)
Notice that 10I is a diagonal matrix, and since Q is orthogonal, the right hand side of
EECS 551: HW 4
Question 1. What are the eigenvalues and eigenvectors of A = xxT + yy T ?
Assume that xT y = = 0
Hint: The desired eigenvectors of A must be in the range of A. Hence they must be linearly related
to x and y.
Check your answers numericall
EECS551: HW3 SOLUTIONS
Problem 1
The transition probability matrix is
Cheese
0
1/2
1/2
Cheese
P =
Grapes
Lettuce
Grapes
4/10
1/10
5/10
Lettuce
6/10
4/10
0
T
Let = 1 2 3 be the equilibrium distribution of the states (Cheese, Grapes, Lettuce), and Pij
i
EECS 551: HW 4 SOLUTIONS
Problem 1
Given A = xxT + yy T .
The rank of A is at most two. It is equal to zero when xxT = yy T and equals one when x is collinear with
y. We now treat the setting where the rank of A is two. In other words, x and y are linearl
EECS 551: Homework 5
Problem 0 (2 pts):
Please fill out the online course evaluation for 551. Once you complete your rating for the course, a
confirmation notice will appear on the web page. Please print this page and submit
it to earn credit for this pro
EECS 551: HW 5 SOLUTIONS
Problem 1
n
T
The idea is to view the image as a matrix A = i=1 i ui vi where in this case n = 480. We know that
k
T
the optimal (with respect to any unitarily invariant norm) rank-k approximation to A is Ak = i=1 i ui vi .
This a
University of Michigan
Fall 2011
EECS551: Practice Problems 2
Instructor: Sandeep Pradhan
1 State TRUE or FALSE by giving reasons. If you give no reason or a wrong reason, you may not
get credit.
The eigen values of matrix
54
25
are 1 and 10.
The eigen
50
Lecture 12
Example 4: Let S be a 2-dimensional space of real vectors. Consider the following inner product
with a parameter such that 0 1:
x, y = [x1 x2 ]
1
1
y1
y2
= x1 y1 + x2 y1 + y2 x1 + x2 y2
Question: Is it a valid inner product ?
Answer: Conditi
58
Lecture 13-15
Subspaces: We have introduced the notion of angle, length and distance into the space of
signals. The next concept we look at is related to the notion of a plane or line in the space of
signals. Note that in 3-dimensions to specify a plan
67
Lecture 16-18
In many practical applications, we are asked to approximate a given signal using a linear
combination of a xed collection of elementary signals. Recall that while studying DT signals,
in the rst part of the course, we looked at at least 3
79
Lecture 19-20
3. Least Squares Filtering: Consider the example of acoustic echo cancellation in teleconferencing applications. Input speech signal f [n] enters the system. It is converted
into an acoustic signal which is radiated by a loudspeaker into
86
Lecture 21
Eigen Values and Eigen Vectors: To study linear systems we will develop the
notion of eigen functions. This notion was used in rst part of the course to introduce
Fourier transforms. Let us look at some examples of linear systems.
1. Let S s
92
Lecture 22-23
The concept of eigen values and eigen vectors is applicable to any Hilbert space and for
any linear transformation. The topic that deals with this concept is called linear operator
theory.
Procedure for nding eigen values and eigen vector
100
Lecture 24-25
Applications of eigen vectors and eigen values:
1. Karhunen-Loeve Transform (KLT): In many image processing applications, one
would like to transmit images (512 x 512) over a noisy channel to a remote receiver.
In such cases the informat
EECS 551: MATRIX METHODS FOR SIGNAL PROCESSING, DATA ANALYSIS & MACHINE LEARNING
EECS 453: APPLIED MATRIX ALGORITHMS FOR SIGNAL PROCESSING, DATA ANALYSIS & MACHINE LEARNING
Summary: Theory and application of matrix methods to signal processing, data analy
EECS 551: MATRIX METHODS FOR SIGNAL PROCESSING, DATA ANALYSIS & MACHINE LEARNING
EECS 453: APPLIED MATRIX ALGORITHMS FOR SIGNAL PROCESSING, DATA ANALYSIS & MACHINE LEARNING
Summary: Theory and application of matrix methods to signal processing, data analy