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.
*

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: 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 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 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 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 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 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 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 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

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.

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

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 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

Chapter 7
Google PageRank
The worlds largest matrix computation. (This chapter is out of date and needs a
major overhaul.)
One of the reasons why GoogleTM is such an eective search engine is the
PageRankTM algorithm developed by Googles founders, Larry Pa

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4:56 PM
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4:57 PM
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Thursday, September 25, 2

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