lecture 39: quick review from previous lecture
TRANSCRIPT
Lecture 39: Quick review from previous lecture
• We have reviewed permuted LU factorization, inverse of a matrix, positive(semi)definite, determinant, solving a linear system and many others.
—————————————————————————————————Today we will review some concepts.
- Lecture will be recorded -
—————————————————————————————————
• Information for Final Exam and Course Grade has been posted on Canvas, see”Announcements”.
MATH 4242-Week 16-1 1 Spring 2020
Problem 14: Find the QR factorization of A =
0
@0 �1 00 0 �32 �1 �3
1
A. Clearly
identify the orthogonal matrix Q and the upper triangular matrix R.
Problem 15:
a) Find the matrix norm of A =
0
@�4 0 00 �1 00 0 7
1
A, with respect to the standard
Euclidean norm kyk2 =py21 + y22 + y23 on R3.
MATH 4242-Week 16-1 3 Spring 2020
Ant 492 493
V. = a , = ( E ) , g ( Y)v. = ar - 9Y' 4=1 :o) . E- ( Ilvs = as - c%Y u
.- ca÷iu= 1%1,95. II)
Then a -
- f ! ! ÷ !=
* l ! "
÷ ) . "it! .is=p:
"
: -41. #
NAH = Fia; tail = 7.
b) Suppose kxk = 3 and kyk = 1. What is the maximum possible value for hx,yi?What relationship must hold between x and y if this value is to be achieved?
c) Suppose hx,yi = 0, kxk = 2, kyk = 1, and z = 2x � 3y. What is hx, zi?What is hy, zi? What is kzk?
Problem 16: Find all vectors in R3 orthogonal to both (1, 2, 0)T and (2, 5, 2)T .
MATH 4242-Week 16-1 4 Spring 2020
( x. x > = 11×112 .
1) / Cx, y> / E 11×1111911.=3
.
maximum valueis 3
'
#2) X =3 Y-
'
#
- o
1) (X. Z > = ( X, 2×-3 y) = (X , 2x> - CX , 377=24×7 -34%= 211×112 = 8
2) Cy,z> = - 3
.
-#
3) 112112 = Cz,z > = CTx-tyx-fip-44.is> - 647%
>O
- 6#t gey.D= 4.22+9=25 .
- - A - ""
VIT X = o - Y K.
viii. o' ( LEE ) x = O .
A①
( o' ? ;) .
I = ( Y) z .
e-R.
Problem 17: Write down the 2-by-2 matrix A satisfying Av1 = w1 andAv2 = w2, where v1 = (1, 1)T , v2 = (�1, 1)T , w1 = (1, 1)T , and w2 = (�2,�2)T .
Problem18: Find a 2-by-2 matrix A with eigenvalues 2 and �3 and correspond-ing eigenvectors (1,�1)T and (1, 0)T .
Problem 19: Write out the SVD of the matrix A =
✓1 �11 �1
◆.
MATH 4242-Week 16-1 5 Spring 2020
JV, -_ Wi
Av, =wz: A sk !
-
- Cw , wa ]
A = Ew, wz ) s -' = If } I ).
#
't.
"vz
AV, = 2 V,
Av. =-3 v,i Alj]=l4 Va ) f} µA- stools - '
= ( III.*
STS = ( I ? ).
o= dee CATA - AI )a = 4
,O
.
=
¥4. (STA- GI) r -
- o, v = EH !
u= A¥= ¥41.
Reduced SD of Ais A- rill) CDC# E).
*.
Problem 20: Find a 2-by-3 matrixA having rank 1 whose singular value is 2, leftsingular vector is u = (1, 2)T/
p5, and right singular vector is v = (1, 0, 1)T/
p2,
that is, Av = 2u.
Problem 21: Suppose A is a 2-by-2 real matrix for which 1�2i is an eigenvalue.Find the trace and determinant of A.
Problem 22: 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.
MATH 4242-Week 16-1 6 Spring 2020
=
A- = Luv 'T.
= Iff) He 0¥= rs
=U[2) VT = ⇐ ( f ) ( I 01)(Reduced SVD)
, o l= ( z o z ). #
I ¥2 i are eigenvalues of A .
det A = D , -- . An = (Hzi ) ( I- zi ) =I#
Trace of A = a" t . - - t Ann = ( Aij )#n
.= D, t Az
= Ctu ) + ( I- zi) =L . #
Operator nom of A : HAH -
- qq.nl " il = IFrobenius norm of A : Il Atf = It
= 5- #
.
Problem 23: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.Find the determinant of A.
Problem 24: Suppose A has characteristic polynomial pA(�) = �2 � 2� + 2.Find the characteristic polynomial of A�1.
Problem 25: Suppose A = AT is a symmetric 2-by-2 matrix, and detA = 6.Suppose that Av = 2v, where v = (1, 1)T . Write an spectral factorization of A.
MATH 4242-Week 16-1 7 Spring 2020
→def A = 17.172=2
. # /v
Pala ) = (R - D . ) ( R - da ) = - - - - t
"doe CA-92)
TE
Pa. . (5) = det ( A " - II )=det( At ( I - IA ) )= det A" dee ( I - E.Al.= detail
"
. det (C- A) (A - ¥2) )deter 't
= doe EAT detCA-EIPs.tl-- Ies -- I
= 'T E k¥5 - ¥+2) = I'- Itt.#
6 -_ dots = 2 - a ⇒ 17=-3.Vz: Eigenvector of 17=3
,
Va is orthogonal to V=
Taking Vz± (t ).
A-- c ' toil'
*
Problem 26: 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 basesof V and W .
Problem 27: 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? Whatare the singular values of AT? What is the determinant of A?
MATH 4242-Week 16-1 8 Spring 2020
P'' '= ( x. I 1 ,
P'''= I *2. x. it
( (x ) = Jot tdt = I X ' t O - X -10.1
L ( 17 = f! Idt = O X't X x o - I
Thus, matrix representation of L is
A- ( ÷.It
. #
→
1) KIA ) = 3,1 =3
4 Sir , of A"
= I. I
'
,'s
3) su, of AT = I,
2,3.
4) dot A = det (UI VT ) = £6( full sup) .
#
TMZ that UTU =UUT= II = det UT dei U = (docu )
'
det U =L ! Jsimilarly , der V = It
.
Problem 28: Suppose A is a matrix with singular values 2, 3 and 8. Supposeu and v are the left and right singular vectors of A with singular value 8, and letB = 8uvT . Find kA� Bk2 and kA� BkF .
Problem 29: Suppose A = uvT , 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 allvectors x that minimize kAx�bk2. Also, find the unique vector x that minimizeskAx� bk2 and has the smallest Euclidean norm.
MATH 4242-Week 16-1 9 Spring 2020
Eu an matrix norm
I = f UVT t 3 Unit 2 Us.
A -B = Susu'T t 2434T .
So, HA-Blk = 3
11 A- BHf= ME =D
.
' l At -_ v." Cnut..= ( ¥
.)
E- At b = ( II ).
.
has the smallest Euclideannorm.
4 All least squaressolution are X*t E
,
where ZE Ker A .
Finding a basis for Ker A :
we knew V=rT( !) is in wings, soE- (4)t is in Ker A .
Then *tz= (E) t.tt ) , THR . #