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 equa
1.1 If A R
and is a scalar, what is det
( A )? What is det (A )?
nn
Solution:
Det (A ) = det( A) .
n
Det (A ) =(
1) det( A) .
n
1.2 If A is orthogonal, what is det A? If A is unitary, what is det A?
EECS 453/551: 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
EECS 551/453 - HOMEWORK 1
Reading pertaining to the problem set: Chapter 1 of Laub
Reading for next week: Chapter 2 Section 9.1 of Laub
EECS 453 students need only attempt problems marked with an astr
EECS 551/453: HW SOLUTIONS
PROBLEM 1 ()
Let e Rm 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
EECS 551/453: HW SOLUTIONS
PROBLEM 1 ()
Here A Rnn have the SVD A = U V T .
"
B=
0
A
AT
0
#
We have that:
"
zI
det(B zI) =
AT
A
zI
#
= det(zI) det(zI AT (zI)1 A)
n
2
T
= (z) det(z I A A)/(z)
By Proper
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 ma
University of Michigan
Fall 2016
EECS 551: Midterm Exam 1
Monday October 24, 2016
Instructions:
Total points for credit = 100.
You may use without rederiving any of the results derived in the class
EECS 453/551
Video background subtraction using SVD
In this exercise we will discover how the SVD can magically estimate the background in a video even
when there is a lot of noise and missing data.
I
EECS 551/453 - Homework Problems
EECS 453 students need only attempt problems marked with an astrisk (*). EECS 551 students must
attempt all problems.
Reading for this week: Chapters 5 and 8.1 of Laub
EECS 551/453 - Homework Problems
EECS 453 students need only attempt problems marked with an astrisk (*). EECS 551 students must
attempt all problems.
Reading pertaining to the problem set: Chapter 1
EECS 453/551: HW 3
Reading for next week: Chapter 2 and Chapter 3 of Laub
EECS 551 students solve all problems. EECS 453 students only solve problems with a *.
*Problem 1. This what is known about the
EECS 453/551: HW 11 SOLUTIONS
*Problem 1
Here is my code for generating the plot in Fig. 1.
load(karate.mat);
figure(1);
set(gcf,color,w);
subplot(221);
spy(A);
subplot(222);
theta = linspace(0,2*pi,3
EECS 551/453 - HOMEWORK 1
Reading pertaining to the problem set: Chapter 1 of Laub
Reading for next week: Chapter 2 Section 9.1 of Laub
Problems marked with an asterisk (*) are for EECS 453
*Problem 1
EECS 453/551: HW 10 SOLUTIONS
Problem 1 (*)
Will be uploaded separately.
Problem 2 (*)
The following was my code for the problem:
clear all;
close all;
clc;
load Q_med.mat % query matrix
load A_med.ma
EECS 453/551: HW 4
EECS 551 students solve all problems. EECS 453 students only solve problems with a *
Problem 1. What are the eigenvalues and eigenvectors of A = xxT + yy T ?
Assume that xT y = = 0
EECS 453/551: HW 3
Reading for next week: Chapter 2 and Chapter 3 of Laub
EECS 551 students solve all problems. EECS 453 students only solve problems with a *.
*Problem 1. This what is known about the
EECS 453/551: HW 6
EECS 453 students only attempt the questions marked with a *
*Problem 1.
If D is an m n diagonal matrix then D is an n m diagonal entries whose non-zero entries are the
recriprocal
EECS 453/551: HW 4
EECS 551 students solve all problems. EECS 453 students only solve problems with a *
Problem 1. What are the eigenvalues and eigenvectors of A = xxT + yy T ?
Assume that xT y = = 0
EECS 453/551: HW 8
Reading: Chapter 6 and Chapter 8 of Laub. EECS 453 students only solve problems marked with a *
Problem 1.
Find the (orthogonal) projection of the vector [2
4]T onto the subspace of
EECS 453/551: 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 wher
EECS 453/551: HW 8
Reading: Chapter 6 and Chapter 8 of Laub. EECS 453 students only solve problems marked with a *
*Problem 1.
Find the (orthogonal) projection of the vector [2
4]T onto the subspace o
EECS 453/551: HW 6 SOLUTIONS
Problem 1 (*)
(A b)T (I A A)y
=bT (A )T (I V U T U V T )y
=bT U ( )T V T (V V T V V T )y
=bT U ( )T ( )T )V T y
If Rmn = diag(1 , 2 , . . . , r ), we can verify that ( )T
EECS 453/551: HW 10 SOLUTIONS
Problem 1 (*)
Dene the diagonal matrix n = diag(n1 , n2 , . . . , 1) so that we can denote
We need to minimize:
n1 n2
n = diag(
,
, . . . , 1).
| n (yn xn h)|2
T
where
Exercises for Vectors, Matrices, and Least Squares
Stephen Boyd
Lieven Vandenberghe
September 23, 2016
This is a collection of exercises for the (draft) book Vectors, Matrices, and Least Squares, by
S
F17 EECS 551
Make-up Homework Set, Due 4PM Thu. Oct. 26
1
Pr. 1.
A symmetric matrix A is positive semi-definite i all of its eigenvalues are non-negative, i.e., i we can write
A = U U T for U orthogon
J. Fessler, October 19, 2017, 18:06 (study version)
6.35
6.3 Subspace learning
This section discusses an application of low-rank matrix approximation to subspace learning. In this setting,
the low-di
F17 EECS 551
Homework 8, Due 4PM Thu. Nov. 9
1
Pr. 1.
(Additivity of nuclear norm)
Let A and B be two matrices of the same size. Because the nuclear norm is a matrix norm, we know (by the triangle
ine
F17 EECS 551
Homework 3, Due 4PM Thu. Sep 28
1
Pr. 1.
Determine how eigenvalues and eigenvectors of
B=
0
AT
A
,
0
are related to singular values and singular vectors of A Rnn . In other words, if A =