Assignment #2
Point Value: 15 points
CS 184: Foundations of Computer Graphics
Fall 2009
Prof. James O’Brien
page 1 of 3
Due Date: Sept. 8th, 5:00 pm
UNIVERSITY OF CALIFORNIA
UNIVERSITY
OF
CALIFORNIA
College of
of this
Engineering,
Department
Electrical
Engineering
and ComputerInSciences
The purpose
assignment
is to test of
your
prerequisite
math background.
general, you should
find this
assignment
to be fairly
easy.Fall
This
must be done
solo.
To Graphics
submit,
please slide unCollege
of Engineering,
Department
ofassignment
Electrical
Engineering
Computer
Sciences
Computer
Science
Division:
2006
CS-184 Foundations
ofand
Computer
der Prof. O’Brien’s door.
Computer Science Division: Professor:
Fall 2006 CS-184
Foundations of Computer Graphics
James O’Brien
Professor:
James O’Brien
Assignment
1
Introduction: Some of these questionsPoint
mayvalue:
look a40
little
difficult. If they do, consider reviewing how dot
Assignment pts
1
products and cross products are formed, how matrices are multiplied with vectors and with each other,
Point value: 40 pts
how determinants are formed and what eigenvalues are. This exercise could take a very long time if
This homework must be done individually. Submission date is Friday, September 8th 2006, at
you don’t think carefully about each problem, but good solutions will use one or two lines per question.
10:00am
under the door
633 individually.
Soda Hall
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Submission
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find very
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are multiplied
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and
of these questions may lookhow
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consider
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other,
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are formed
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This exercise
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ing how
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matrices areare.
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previous years. To be honest, if you need to cheat on this assignment then just go drop the class
now.Question 1: Two vectors in the plane, i and j, have the following properties (i · i means the
dot Question
product between
i and
i): i in
· i the
= 1,plane,
i · j = i0,and
j · jj,=have
1. the following properties (i · i means the
1: Two
vectors
dot product between i and i): i · i = 1, i · j = 0, j · j = 1.
1. Is there a vector k, that is not equal to i, such that: k · k = 1, k · j = 0? What is it? Are
many
vectors
1. there
Is there
a vector
k, with
that these
is notproperties?
equal to i, such that: k · k = 1, k · j = 0? What is it? Are
many
vectors
with that:
these kproperties?
2. there
Is there
a vector
k such
· k = 1, k · j = 0, k · i = 0? Why not?
2. If
Is ithere
vector
k suchinthat:
k · kwould
= 1, kthe
·j=
0, k · i to
= the
0? Why
3.
and ja were
vectors
3D, how
answers
abovenot?
questions change?
3. If i and j were vectors in 3D, how would the answers to the above questions change?
Question 2: For three points on the plane (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) show that the determinant
of
Question
2: For three points on theplane (x1 , y1
), (x2 , y2 ) and (x3 , y3 ) show that the deterx1 y1 1
minant of




x12 yy12 1
 x
1 
 x3 y3 1 
 x2 y2 1 
x3 ycorners
3 1
is proportional to the area of the triangle whose
are the three
a
line, to
what
theofvalue
of the determinant?
thisthree
give
is straight
proportional
theisarea
the triangle
whose cornersDoes
are the
three
points
lie
on
a
line?
Why
do
you
think
so?
a straight line, what is the value of the determinant? Does this give
points. If these points lie on
a usefulIftest
to points
tell whether
points.
these
lie on
a useful test to tell whether
three
points
lie
on
a
line?
Why
do
you
think
so?
Question 3: The equation of a line in the plane is ax + by + c = 0. Given two points on the
plane,
show how3:toThe
findequation
the values
a, b,inc for
lineisthat
thosetwo
twopoints
points.
Question
of of
a line
the the
plane
ax +passes
by + cthrough
= 0. Given
onYou
the
may
find
the
answer
to
question
2
useful
here.
plane, show how to find the values of a, b, c for the line that passes through those two points. You
mayQuestion
find the answer
useful
4: Lettoequestion
e2 =here.
(0, 1, 0), e3 = (0, 0, 1). Show that if {i, j, k} is {1, 2, 3},
1 = (1, 0,20),
{2, 3,
1},
or
{3,
1,
2},
then
e
×
e
=
e
,
where
is e
the
cross product. Now show that if {i, j, k} is
Question 4: Let e1 =i (1, 0,j 0), ek2 = (0, 1,×0),
3 = (0, 0, 1). Show that if {i, j, k} is {1, 2, 3},
{1, 3,
3, 1},
2}, {3,
2, 1}
or {2,
1, 3},
ej = −e×k .is the cross product. Now show that if {i, j, k} is
{2,
or {3,
1, 2},
then
ei ×then
ej =eie×
k , where
{1, 3, 2}, {3, 2, 1} or {2, 1, 3}, then ei × ej = −ek .
Question 2: For three points on the plane (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) show that the determinant
of
Question
2: For three points on theplane (x1 , y1
), (x2 , y2 ) and (x3 , y3 ) show that the determinant of
Assignment #2
x1 y1 1 



 x12 y12 1 

 x Foundations
CS 184:
of Computer Graphics
 23 y23 1 
page 2 of 3
Fall 2009
Point Value: 15 points
x3 ycorners
3 1
is proportional to the area of the triangle whose
are the three points. If these points lie on
Prof.
James
O’Brien
Due
Date:
Sept.
8th,
5:00
pm
a straight
line, to
what
theofvalue
of the determinant?
thisthree
give points.
a usefulIftest
to points
tell whether
is
proportional
theisarea
the triangle
whose cornersDoes
are the
these
lie on
points
lie what
on a line?
so?
athree
straight
line,
is theWhy
valuedoofyou
thethink
determinant?
Does this give a useful test to tell whether
three
points lie 3:
on The
a line?
Why do
think
so?plane is ax + by + c = 0. Given two points on the
Question
equation
of you
a line
in the
plane,
show how3:toThe
findequation
the values
a, b,inc for
lineisthat
thosetwo
twopoints
points.
Question
of of
a line
the the
plane
ax +passes
by + cthrough
= 0. Given
onYou
the
may
find
the
answer
to
question
2
useful
here.
plane, show how to find the values of a, b, c for the line that passes through those two points. You
mayQuestion
find the answer
useful
4: Lettoequestion
= (1, 0,20),
e =here.
(0, 1, 0), e = (0, 0, 1). Show that if {i, j, k} is {1, 2, 3},
1
2
3
{2, 3,
1}, or {3, 1,
ei (1,
× e0,j 0),
= eek2, where
is ethe
j,2,
k}3},
is
Question
4:2},
Letthen
e1 =
= (0, 1,×0),
(0, 0,product.
1). ShowNow
thatshow
if {i,that
j, k}ifis{i,
{1,
3 =cross
{1, 3, 1},
2}, {3,
2, 1}
or {2,
1, 3},
{2,
or {3,
1, 2},
then
ei ×then
ej =eie×
×k .is the cross product. Now show that if {i, j, k} is
j = −e
k , ewhere
{1, 3, 2}, {3, 2, 1} or {2, 1, 3}, then ei × ej = −ek .
Question 5: For four points in space (x1 , y1 , z1 ), (x2 , y2 , z2 ), (x3 , y3 , z3 ), (x4 , y4 , z4 ) show that
the Question
determinant5:ofFor four points in space
 (x1 , y1 , z1 ), (x2 ,y2 , z2 ), (x3 , y3 , z3 ), (x4 , y4 , z4 ) show that
x1 x2 x3 x4 , y , z ), (x , y , z ), (x , y , z ) show that
Question
5:
For
four
points
in
space
2 2 2
3 3 3
4 4 4
the determinant of

 y(x1 ,yy1 , zy1 ), (x
y44 ,

y2 , z2 ), (x3 , y3 , z3 ), (x4 , y4 , z4 ) show that
11 1 x
x
, y221 , zx133), (x
the Question
determinant5:ofFor four points in space

2
 x(x


 x

yz11 x
yz22 x
yz33 x
yz44 




 x
1
1
1
1
yz11 x
yz22 x
yz33 x
yz44 






y
y
y
y


z
z
z
z


1
2
3
4
1
1
1
1
is proportional to the volume of the tetrahedron
whose
corners
are the four points. If these points




z
z
z
z
1
1
1
1
1
2
3
4
lie on a plane, what is the value of the determinant? Does this give a useful test to tell whether
the determinant of
is proportional to the volume of the tetrahedron
1 think
1 whose
1 so?1corners are the four points. If these points
four
points
a plane?
Why
dodeterminant?
you
is
to lie
theon
of the
tetrahedron
whose
corners
thea four
points.
If tell
these
points
lieproportional
ongiven
a plane,
what
isvolume
the
value
of the
Does
thisare
give
useful
test to
whether
is
proportional
to lie
the
of the
tetrahedron
whose
corners
the
If tell
these
points
lie
on
a plane,
what
isvolume
the
value
ofathe
Does
this
give
useful
test to
whether
four
given
points
on
a plane?
Why
dodeterminant?
you
think
so?ax
Question
6:
The
equation
of
plane
in space
is
+ by
+are
cz
+ da four
=
0. points.
Given
three
points
in
lie
on
a
plane,
what
is
the
value
of
the
determinant?
Does
this
give
a
useful
test
to
tell
whether
four
given
points
lie
on
a
plane?
Why
do
you
think
so?
space,
show
how
to
find
the
values
of
a,
b,
c,
d
for
the
plane
that
passes
through
those
three
points.
Question
6:lieThe
of
a plane
in space
+ by + cz + d = 0. Given three points in
four
givenfind
points
on equation
a plane?
Why
you
think is
so?ax
You
may
the
to values
question
4do
useful
here.
Question
6:toanswer
The
of
a a,
plane
ax + that
by + passes
cz + d through
= 0. Given
three
points
in
space,
show how
findequation
the
of
b, c,ind space
for theisplane
those
three
points.
Question
6:
The
equation
of
a
plane
in
space
is
ax
+
by
+
cz
+
d
=
0.
Given
three
points
in
space,
show
how
to
find
the
values
of
a,
b,
c,
d
for
the
plane
that
passes
through
those
three
points.
7: answer
Let p0 , to
p1question
, p2 be three
distinct
YouQuestion
may find the
4 useful
here.points in space. Now consider the cross product
space,
show
how
to
find
the
values
of
a,
b,
c,
d
for
the
plane
that
passes
through
those
three
points.
You
may
find
the
answer
to
question
4
useful
here.
n =Question
(p0 − p1 ) ×7:(pLet
p2,).pWhat
does
thisdistinct
vector points
mean geometrically?
Let p be the
anycross
pointproduct
on the
0 −p
, p2 be
three
in space. Now consider
0 to 1question
You
may
find
the
answer
4
useful
here.
plane
by7:p0Let
, p1pand
isthis
the
geometric
relationship
between
p−
p0 (look
at
, p1p, 2p;2what
be
three
distinct
points
in space. Now
consider
cross
product
n
=Question
(pformed
does
vector
mean
geometrically?
Let n
p and
be the
any
point
on the
0 − p1 ) × (p0 − p02 ). What
the
dot
product)?
Show
that,
if
p
=
(x,
y,
z)
the
equation
of
the
plane
must
be
n.(p
−
p
)
=
0.
Question
7:
Let
p
,
p
,
p
be
three
distinct
points
in
space.
Now
consider
the
cross
product
n
=
(p
−
p
)
×
(p
−
).
What
does
this
vector
mean
geometrically?
Let
p
be
any
point
on
the
0
1p ; 2what is the geometric relationship between n and p − p (look at
0
1 by p00, p1 0
2
plane formed
and
2
0
n
=
(p
−
p
)
×
(p
−
p
).
What
mean
geometrically?
Lettranspose
pbeand
ben.(p
any
on0.the
plane
formed
by
p
,
p
and
p2if; what
thez)vector
geometric
relationship
between
n
p exist,
−
p
at
0
1
0
2
0
1
8: Suppose
that
A
aisthis
square
matrix,
and
itsthe
inverse
and
and
are
the Question
dot product)?
Show that,
p does
=is (x,
y,
the
equation
of
plane
must
−point
p00 )(look
=
plane
formed
by
p
,
p
and
p
;
what
is
the
geometric
relationship
between
n
and
p
−
p
(look
at
the
dot
product)?
Show
that,
if
p
=
(x,
y,
z)
the
equation
of
the
plane
must
be
n.(p
)
=
0.
0
1
2
0
equal;
a matrix 8:
with
these properties
called an
orthonormal
matrix.and
Show
that, forexist,
any0 angle
θ,
Question
Suppose
that
A
is (x,
ais square
matrix,
and of
itsthe
inverse
transpose
and
are
the
dot
product)?
Show
that,
if
p
=
y,
z)
the
equation
plane
must
be
n.(p
−
p
)
=
0.
0
the Question
matrix M(θ),
defined
by
the
array
below,
is
orthonormal.
Suppose
that A is ais square
matrix,
and its inverse
transpose
and are
equal; a matrix 8:
with
these properties
called an
orthonormal
matrix.and
Show
that, forexist,
any angle
θ,
Question
8:
Suppose
that
A
is
a
square
matrix,
and
its
inverse
and
transpose
exist,
and are
equal;
a
matrix
with
these
properties
is
called
an
orthonormal
matrix.
Show
that,
for
any
angle
θ,
'
(
the matrix M(θ), defined by the array below, is orthonormal.
cosis
θ orthonormal.
sin θ
equal;
a matrix
with
theseby
properties
called
an
orthonormal
matrix. Show that, for any angle θ,
the
matrix
M(θ),
defined
the arrayisbelow,
' − sin θ cos θ (
the matrix M(θ), defined by the array below,
' cosisθ orthonormal.
sin θ (
θ
sin θθ (
' −cos
sin θ cos
Show also that M(θ1 + θ2 ) = M(θ1 )M(θ2 );
use
sintoθ argue that the inverse of M(θ) is M(−θ).
−cos
sinθθthis
cos
Finally, for the point p = (0, 1), what geometrical
figure
is given by taking the points given by
− use
sin θthis
cos
Show also that M(θ1 + θ2 ) = M(θ1 )M(θ2 );
toθ argue that the inverse of M(θ) is M(−θ).
M(θ)p
for
every
possible
value
of
θ?
Show also
= M(θ
to argue
that the
inverse the
of M(θ)
M(−θ).
1+
2 ) (0,
1 )M(θ
2 ); use this figure
Finally,
forthat
the M(θ
point
p θ=
1), what
geometrical
is given
by taking
pointsis given
by
Show
also
that
M(θ
+
θ
)
=
M(θ
)M(θ
);
use
this
to
argue
that
the
inverse
of
M(θ)
is
M(−θ).
Finally,
for
the
point
p
=
(0,
1),
what
geometrical
figure
is
given
by
taking
the
points
given
by
1
2
1
2
Question
You arevalue
givenoffour
M(θ)p
for every9:possible
θ? vectors in the plane, x1 and x2 , b1 and b2 , and you are told
Finally,
for
the
point
p
=
(0,
1),
what
geometrical
figure
is
given
by
taking
the
points
given
M(θ)p
for
every
possible
value
of
θ?
that there is a matrix M such that Mx1 = b1 and Mx2 = b2 . Now if you are given a vector xby
3,
Question
9:possible
You arevalue
givenoffour
vectors in the plane, x1 and x2 , b1 and b2 , and you are told
M(θ)p
for
every
θ?
howQuestion
do you determine
? (hint:
howin
tothe
findplane,
out what
M is
the data,
andyou
then
3given
x2from
, b1 and
b2 , and
areapply
told
9: YouMx
aresuch
fourshow
vectors
x1 and
that there is a matrix
M
that
Mx
1 = b1 and Mx2 = b2 . Now if you are given a vector x3 ,
it
toQuestion
x3 ). is a matrix
and
x
,
b
and
b
,
and
you
are
told
9:
You M
aresuch
giventhat
fourMx
vectors
in
the
plane,
x
that
there
=
b
and
Mx
=
b
.
Now
if
you
are
given
a
vector
x3 ,
12
1
2
how do you determine Mx3 ? (hint: show1 how 1to find out2 what
M is 2from
the data,
and then apply
that
there
is
a
matrix
M
such
that
Mx
=
b
and
Mx
=
b
.
Now
if
you
are
given
a
vector
x3 ,
how
do
you determine
Mxdown
show1 how 1toequation
find out2ofwhat
M is from
then
apply
3 ? (hint:
10: Write
the parametric
the2 points
on athe
linedata,
that and
passes
through
it toQuestion
x3 ).
how
do
youin
determine
show how to find out what M is from the data, and then apply
it
topoints
x3 ).
3 ?p(hint:
and
two
space, p1Mx
2 ; now write down the parametric equation of the points on a plane
Question
10:
Write
down
the
parametric
equation of the points on a line that passes through
it
to
x
).
3
that contains
these
points
and aparametric
third point, p3 . Show
howpoints
to useonthese
parametric
equations
10: two
Write
down
of the
line
that
passes
p2 ; the
now write downequation
the parametric
equationa of
the
points
on through
a plane
twoQuestion
points in space,
p1 and
andQuestion
some inequalities
the
todown
specify
(a) all
points
onathe
line
that
lie
between
10: Write
down
equation
of the points
on
line
that
passes
through
pparameters
; the
nowaparametric
write
the
equation
of
the
points
on
a plane
two
in these
space,
p1onand
2 and
that points
contains
two
points
third point,
p3 . parametric
Show how to
use these
parametric
equations
p1 and
p2 and
(b) all
points
onathe
triangle
vertices
p1 , these
p2 of
and
p3points
.
and
p2 and
; now
write
down whose
the
equation
the
on
a plane
two
points
in these
space,
p1the
that
contains
two
points
third
point,
p3 . parametric
Show
howare
to
use
parametric
equations
and some inequalities on the parameters to specify (a) all the points on the line that lie between
that
contains
these
two
and a third
p3 .(a)
Show
how
to useon
these
equations
andQuestion
some
inequalities
on
parameters
topoint,
specify
allhas
the
points
the parametric
line
between
and λlie
11: all
Mthe
ispoints
athe
symmetric
2x2
matrix.
It
two
λ03 .that
1.
p1 and p2 and (b)
points
on thereal
triangle
whose
vertices
areeigenvalues,
p1 , p2 and p
and
some
on the
parameters
to specify
(a) vertices
all the points
p1 and
p2 inequalities
and (b) all the
points
on the triangle
whose
are p1 ,on
p2the
andline
p3 .that lie between
Question
11: all
M is apoints
symmetric
real
2x2 matrix.
It has two
λ03 .and λ1 .
p1 and
(b)
on the
triangle
whose
vertices
areeigenvalues,
p
, px2T and
T
2 and
•Question
If p
both
eigenvalues
positive,
show
for
any 2D
vector
0 1≤
Mx p
≤0 (max(λ
and λ1 .0 , λ1 ))x x.
11: Mthe
isare
a symmetric
realthat,
2x2 matrix.
It has
twox,eigenvalues,
λ
Question 11: M is a symmetric real 2x2 matrix. It has two eigenvalues, λ0 and λ1 .
the Question
dot product)?
Show that,
if A
p =is (x,
y, z) the
equation
plane must
be n.(p exist,
− p )and
= 0.are
Suppose
that
ais square
matrix,
and of
itsthe
inverse
transpose
equal;
a matrix 8:
with
these properties
called an
orthonormal
matrix.and
Show
that, for any0 angle
θ,
equal;
a
matrix
with
these
properties
is
called
an
orthonormal
matrix.
Show
that,
for
any
angle
θ,
8: defined
Supposebythat
is a below,
square is
matrix,
and its inverse and transpose exist, and are
the Question
matrix M(θ),
the A
array
orthonormal.
the
matrix
M(θ),
defined
by
the
array
below,
is
orthonormal.
equal; a matrix with these properties is called
an orthonormal
matrix. Show that, for any angle θ,
'
(
' cosisθ orthonormal.
(
sin
θ
the matrix M(θ), defined by the arrayCS
below,
184:
Foundations
of Computer Graphics
page 3 of 3
cos
θθ cos
sin θθ
−
sin
'
(
Fall 2009
Point Value: 15 points
−cos
sinθθ cos
sin θθ
Prof.
James
O’Brien
Show
also that
M(θ
θ2 ) pm
= M(θ1 )M(θ2 );
toθ argue that the inverse of M(θ) is M(−θ).
− use
sin θthis
cos
Due Date:
Sept.
8th,
5:00
1+
Show
also
that
M(θ
+
θ
)
=
M(θ
)M(θ
);
use
this
to argue
that the
inverse the
of M(θ)
M(−θ).
1 p =
2 (0, 1), what
1
2
Finally, for the point
geometrical
figure
is given
by taking
pointsis given
by
Finally,
for
the
point
p
=
(0,
1),
what
geometrical
figure
is
given
by
taking
the
points
given
by
Show also
) = M(θ
M(θ)p
for that
everyM(θ
possible
of θ?
1 + θ2value
1 )M(θ2 ); use this to argue that the inverse of M(θ) is M(−θ).
M(θ)p
for
every
possible
value
of
θ?
Finally, for the point p = (0, 1), what geometrical figure is given by taking the points given by
Question 9: You are given four vectors in the plane, x1 and x2 , b1 and b2 , and you are told
M(θ)p
for every9:possible
offour
θ? vectors in the plane, x1 and x2 , b1 and b2 , and you are told
You M
arevalue
giventhat
thatQuestion
there is a matrix
such
Mx1 = b1 and Mx2 = b2 . Now if you are given a vector x3 ,
thatQuestion
there
a matrix
M
that
Mx
= bin
Mx
=x1b2and
.MNow
a then
vector
x3 ,
1 how
1toand
x2from
, ifb1you
and
b2given
, and
areapply
told
9: YouMx
aresuch
given
four
vectors
the
how
do
youisdetermine
?
(hint:
show
findplane,
out2 what
is
theare
data,
andyou
3
how
do
you
determine
Mx
?
(hint:
show
how
to
find
out
what
M
is
from
the
data,
and
then
apply
3
that
that Mx1 = b1 and Mx2 = b2 . Now if you are given a vector x3 ,
it
to there
x3 ). is a matrix M such
it
to
x
).
3
howQuestion
do you determine
Mxdown
show how toequation
find outofwhat
M is from
then
apply
3 ? (hint:
10: Write
the parametric
the points
on athe
linedata,
that and
passes
through
it
to
x
).
Question
10:
Write
down
the
parametric
equation
of
the
points
on
a
line
that
passes
through
3
two points
in space, p1 and p2 ; now write down the parametric equation of the points on a plane
p2 and
; the
nowaparametric
write
downequation
the
equation
the
points
on
a plane
two
points
in these
space,
p1 and
10: two
Write
down
of the
a of
line
that
passes
through
thatQuestion
contains
points
third point,
p3 . parametric
Show
howpoints
to
useonthese
parametric
equations
that
contains
these
two
points
and
a
third
point,
p
.
Show
how
to
use
these
parametric
equations
3
the(a)
parametric
equation
of the
onbetween
a plane
two
in space, p1onand
and points
some inequalities
thepparameters
specify
all the points
on the
linepoints
that lie
2 ; now writetodown
and
some
inequalities
on
the
parameters
to
specify
(a)
all
the
points
on
the
line
that
lie
between
that
contains
these
two
points
and
a
third
point,
p
.
Show
how
to
use
these
parametric
equations
p1 and p2 and (b) all the points on the triangle whose
vertices are p1 , p2 and p3 .
3
p
p2 inequalities
and (b) all the
points
on the triangle
whose
vertices
are p1 ,on
p2the
andline
p3 .that lie between
1 and
and
some
on
the
parameters
to
specify
(a)
allhas
thetwo
points
Question 11: M is a symmetric real 2x2 matrix. It
eigenvalues,
λ0 and λ1 .
p1 and
p2 and (b)
on thereal
triangle
whose vertices
areeigenvalues,
p1 , p2 and p
Question
11: all
Mthe
is apoints
symmetric
2x2 matrix.
It has two
λ03 .and λ1 .
Assignment #2
and λ1 .0 , λ1 ))xT x.
11: M isare
a symmetric
realthat,
2x2 matrix.
It has
twox,eigenvalues,
•Question
If both eigenvalues
positive, show
for any 2D
vector
0 ≤ xT Mx λ
≤0 (max(λ
• If both eigenvalues are positive, show that, for any 2D vector x, 0 ≤ xT Mx ≤ (max(λ0 , λ1 ))xT x.
•
one eigenvalue
zero,
show that
vector
x $=x,00such
= 0. , λ ))xT x.
• If
If both
eigenvaluesis
positive,
showthere
that, is
forsome
any 2D
vector
≤ xTthat
Mx Mx
≤ (max(λ
•
If
one eigenvalue
isare
zero,
show that
there
is
some
vector
x $= 0 such
that
Mx
= 0. 0 1
•
one eigenvalue
eigenvalue is
is zero,
positive
and
onethere
is negative,
is that
someMx
vector
$= 0 such
•• If
If
one
show
that
is some show
vectorthat
x $=there
0 such
= 0.x
If
one
eigenvalue
is
positive
and
one
is
negative,
show
that
there
is
some
vector
x
$= 0 such
T
that xT Mx = 0
that
Mx = 0 is positive and one is negative, show that there is some vector x $= 0 such
• If
onex eigenvalue
• A
matrix
definite if, for any vector x, xTT Mx > 0. Assume that M is
T
that
x MxM
= is
0 positive
• symmetric,
A matrix
M
is
positive
definite
if, What
for any
x, about
x Mxits>eigenvalues?
0. Assume that M is
real, and positive
definite.
canvector
you say
real,
and positive
definite.
canvector
you say
• symmetric,
A matrix M
is positive
definite
if, What
for any
x, about
xT Mxits>eigenvalues?
0. Assume that M is
symmetric, real, and positive definite. What can you say about its eigenvalues?
Question 12: M is a 2x2 matrix.
• Show that the eigenvalues of M are the roots of the equation λ2 − trace(M)λ + det(M) = 0.
Something Optional: +5 pts
If you decide to continue into graduate school, you will most likely have to write papers about
your graduate research work. If these papers contain any math, you will need to set your equations
using a tool like LATEX, Mathematica, or MS Word’s Equation Editor. (Listed in order of my own
preference.)
If you use one of these tools (or something similar) to write up the solutions to at least 4
questions, you will get two extra credit points on this assignment. If you do the entire assignment
with the tool you will get a total of five points extra.
My personal preference is for LATEXbecause it produces the nicest looking output. LATEX’s
downside is that it is hard to learn, although once learnt it is fast to use. Mathematica and Word