let m n .suppose haspositivesingular o r u1,,ur 1 r l tired i
TRANSCRIPT
Lecture 39: Quick review from previous lecture
• LetA 2 Mm⇥n (m⇥n real matrices) of rank r. SupposeA has positive singular
values �1 � . . . � �r and corresponding right singular vectors v1, . . . ,vr and
left singular vectors u1, . . . ,ur. Then
A = �1u1vT1 + . . . + �rurv
Tr .
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Today we will
• review some concepts
- Lecture will be recorded -
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• Exam 3: 12/16 (Wednesday) in lecture.
• Practice Exam is on Canvas now.
• Additional o�ce hours next Tuesday from 10 am-11:50 am.
MATH 4242-Week 15-1 1 Fall 2020
-O
- -
= " ' --
I' l" I Tired;:S,
( # 9 solution is modified)
Problem 12: Find a 2-by-2 matrix A with eigenvalues 2 and �3 and corre-
sponding eigenvectors (1,�1)Tand (1, 0)T .
Problem 13: Find a 2-by-3 matrix having rank 1 whose singular value is 2, left
singular vector is (1, 2)T/p5, and right singular vector is (1, 0, 1)T/
p2.
[We have discussed in Lecture 38.]
MATH 4242-Week 15-1 2 Fall 2020
-
I, E
.
A [ v, Va ) = ( 2v, -34 )= [ v, 47 f! go ]
A = Cv. v. If ! 1) fu, rift= C - so I
- #
Problem 14: Write out the SVD of the matrix A =
✓1 �1
1 �1
◆.
MATH 4242-Week 15-1 3 Fall 2020
ATA = (I-
I ) has eigenvalues 4 , O
singular value ,T1¥ = ATA - 41 = (II ) .
v. = (tyrant= ATA = (II) . " = ( th → Kera
.
I. :
Kern cokerAwut
u,= Aft = l !)g, → ing A
Find orthonormal basis for askers :
uz t u , ⇒ u- = (th.
A- Cu . a.) 17151 III. ] (Tulisa)= f. u . ] 12 ] I v.
T ) (Reduced
= f " hi Cyr.-ya ,
suo)-
A- = 2 u , v.T
Problem 15: Suppose A is a 2-by-2 symmetric matrix with eigenvalues 3 and
�4. Find the operator norm of A and the Frobenius norm of A.
Problem 16: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.
Find the determinant of A and the trace of A.
Problem 17: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.
Find the characteristic polynomial of A�1.
MATH 4242-Week 15-1 4 Fall 2020
( T 4,02=3 )-O
enable)- Tnatuenatnx norm !
HAIL = max { I dial,la . I / = 4
HAH#
= Fit = 5. #
A has eigenvalues a, b .Ps (A) = D
'- za +2 -
detA=x ,- -- an
I.' 'II' c'a''I.7. + as far.net.in .
doc A- = ab =③⇒ doestrot) = atb = 2
. #
deeks) det ( A" - az )=doelA④#
=A_ der ( A" ( I - AA ) )= dot ( RA" ( II - Al )( doe l -as' ca - in) ¥157,= diet (ILA
-t ) det ( A -⑤z )± -45 @etA'' ) (Hi - 4kHz)
-742= 172 - R t Yz
.
Problem 18: Suppose A = ATis a symmetric 2-by-2 matrix, and detA = 6.
Suppose that Av = 2v, where v = (1, 1)T . Write the spectral factorization of A.
Problem 19: Let V = P (1)be the space of polynomials of degree 1, andW =
P (2)be the space of polynomials of degree 2. Let L[p](x) =
R x0 p(t)dt denote the
integration operator. Find the matrix representation of L in the monomial bases
of V and W .
MATH 4242-Week 15-1 5 Fall 2020
-
D , .V,= like A = QDQT -- QDQT
⇐ dot A = 2 . Az ⇒ a, =3where Iis orthogonal
since A-AT,ht v
,⇒ V.= ( IKE
.
A -- %.
13519% .
'' '
*→orthogonal
Li p '' '→ plz )-1X
, if I x? x, i)
( ( att b ) = a L Cx ) t BLE 17= [ LEXI Lin ) ( f )
• Lfx ) = f! it dt = II → (kg )• LCM = f! I dt = ×
→
fog )Lll :) ] -- f ! Ill :)
Problem 20: Suppose A is a 3-by-3 matrix with singular values 1,2, and 3.
What is the condition number of A? What are the singular values of A�1? What
are the singular values of AT? What is the determinant of A?
Problem 21: Suppose A is a 3-by-3 symmetric matrix of the form
A = 2u1uT1 � 3u2u
T2 + 9u3u
T3 ,
where u1,u2,u3 2 Rnare nonzero column vector and are orthonormal. What is
the condition number of A? What are the singular values of A�1? What are the
singular values of AT? What is the determinant of A?
MATH 4242-Week 15-1 6 Fall 2020
9 On T,
- -
1) KIA) =%, = 3
→ x. x. x. . then!::II÷,3) I,2, 3 = It 6 It
41 If =E6_
-(spectral factorized
=Cu , um,] (Z-3 g) fu.u.u.TT•
T, = 9 , 02=3 ,03=2 (Ti = lait
t) KCA ) = 9/2=4.5 since A- AT)2) Ya
,43
,42
3)9 , 3,2
4) dot A = 2C -3) g = -54-#
Problem 22: Suppose A is a matrix with singular values 2, 3 and 8. Suppose
u and v are the left and right singular vectors of A with singular value 8, and let
B = 8uvT. Find kA� Bk2 and kA� BkF .
Problem 23: SupposeA = 2uvT, where u = (1,�1)
T/p2 and v = (1, 1)T/
p2.
Let b = (1, 0)T . Find all least squares solutions to Ax = b. That is, find all
vectors x that minimize kAx�bk2. Also, find the unique vector x that minimizes
kAx� bk2 and has the smallest Euclidean norm.
MATH 4242-Week 15-1 7 Fall 2020
A- = 8 UVT t 3 U.zv.tt 2 Us VstB = SEUVT : Best rankle approximation ofA .
[ = 8 UVTT 3 cravat : Best rank 2 s
-
A -B = 3 Uz VT t 2434T has singularvalues 3
,2
.
HA - 13112 = 3 ; HA -BH,-_TE
O=D . #
-
-
-" A = [ u ] 127 CUT (Reduced
At = C v ) ( Yz ] HI
xx = Atb = ( YT.).
4 Ft Ikerd : 2-= (t, )t ,orthogonal-
=fY¥ ) t (f) t , te EIR .
Problem 24: Find the general solutions x(t) = (x1(t), x2(t), x3(t))T to the
following system of di↵erential equations:
8<
:
x01 = 4x1 + x3x02 = 2x1 + 3x2 + 2x3x03 = x1 + 4x3
.
Here xi = xi(t) is a di↵erentiable function of variable t and x0i(t) =ddtxi(t), i =
1, 2, 3. Simplify your answer.
MATH 4242-Week 15-1 8 Fall 2020
A- ( & §! ) ,D= 3
,
3,5
.
⇒ .
v. = ( Y) , h= (g). v. = ( fl
A-- V D V"
nth D= ( { II )✓ = fu, v. is]
Then X'
= Ax= VDV
-"x
⇒ ( y )'
= Dy d key -_V'x
⇒ s=l÷÷÷,Then x=vy= (
"' e:e¥EF¥c.e't*GET . #