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

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 551/453 - 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 astrisk (*). EECS 551 students must
attempt all problems.
*

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 distribution of the states (Cheese, Grapes, Lettuce),

EECS 453/551 - 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
EECS 551 students solve all problems. EECS 453 students solve ONLY the problems that are NOT
marked

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?
Solution:
(1) If A is orthogonal, det A is +1-1.
or
det

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
Hint: The desired eigenvectors of A must be in the ra

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 dietary habits of the mythical Michigan Wolverine who

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 of the non-zero diagonal entries of D. D is referred to

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.mat % term-document matrix
load dict_med.mat % dictionary

EECS 453/551 - 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
EECS 551 students solve all problems. EECS 453 students solve ONLY the problems that are NOT
marked

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 dietary habits of the mythical Michigan Wolverine who

EECS 453/551 - 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
EECS 551 students solve all problems. EECS 453 students solve ONLY the problems that are NOT
marked

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
Hint: The desired eigenvectors of A must be in the ra

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. Express the n m matrix A whose j th row equals j as a

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
Stephen Boyd and Lieven Vandenberghe. They are used in E

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 R3 spanned by the plane
3
3x y + 2z = 0.
*Problem 2.
C

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 where the rank of A is two. In other words, x and y are lin

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 = ( )T through direct multiplication, so
that (A b)T (I

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 yn = y(n) y(n 1) . . .
minimizing:
T
and xn = x(n) x(n

S w w ~ f W @0l f c U A U QW 1Qu U c x w ux ~ U w `w @x VS u w u U Y w w u x w u %S 1u P w gS S W Q1w Y u dc x S vt 3 Y ux w y w w Y U S w w u t gS w W 1g S 1u ` g u vVS w PU QW ow Xq1vd w u S a y wu QW @x U QW ow w qS x vt X ~ lg VS w Y u A S U `@w `S w

Section 5.2 8 Suppose A = uv T is a column times a row (a rank-1 matrix). (a) By multiplying A times u, show that u is an eigenvector. What is ? Solution. Au = (uv T )u = u(v T u) = (v T u) u, so by denition, u is an eigenvector with the associated eigenv

The PageRank Citation Ranking:
Bringing Order to the Web
January 29, 1998
Abstract
The importance of a Web page is an inherently subjective matter, which depends on the
readers interests, knowledge and attitudes. But there is still much that can be said o

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.
Instructions:
1. Logon to Canvas and download the backgr

Thursday, October 2, 2014
9:43 PM
new Page 1
Thursday, October 2, 2014
10:03 PM
new Page 2
Thursday, October 2, 2014
10:13 PM
new Page 3
Thursday, October 1, 2015
4:56 PM
new Page 4
Thursday, September 25, 2014
4:57 PM
new Page 5
Thursday, September 25, 2

Cleves
Corner
Professor SVD
By Cleve Moler
Stanford computer science professor Gene Golub has done more than anyone to
make the singular value decomposition one of the most powerful and widely used
tools in modern matrix computation.
from its SVD. Take 1

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