DSC 155 & MATH 182: “WINTER” 2020HOMEWORK 3

Available Friday, February 28 Due Friday, March 6

Turn in the homework on Gradescope before midnight. Late homework will not be ac-cepted.

1. Let P be an orthogonal projection matrix, with column space V . Show that each eigen-value of P is either 0 or 1. What can you say about eigenvectors of of P , in relation toV ?

2. For 1 i, j N , let Eij denote the N ⇥N matrix with a 1 in the (i, j) entry and zeroeselsewhere: [Eij]ab = �ia�jb. The {Eij}1i,jN are called the matrix units.

(a) Show that EijEk` = �jkEi`.(b) Show that, for any matrix X 2 MN⇥N , Tr(XEij) = [X]ji.(c) Use the matrix units to prove the following: if X is an N ⇥ N matrix with the

property thatTr(XAB) = Tr(XBA) for all A,B 2 MN⇥N

then X = �IN for some constant � 2 R. [Hint: apply the identity with A andB various choices of matrix units, and use parts (a) and (b) to conclude that theoff-diagonal entries of X are 0, while the diagonal entries are all equal.]

3. Let C+ = {z = x + iy : y > 0} be the upper half-plane, and consider the functionG : C+ ! C defined by

G(z) =�z +

pz2 � 4

2.

The function G is the Stieltjes transform of a probability density: compute this density.[Hint: Don’t write things out in terms of z = x + iy; just assume the complex squareroot works the way you would hope.]

Problems 4 and 5 are about the Marchenko–Pastur density f%, which is defined as follows.For % > 0, set

a(%) = (1�p%)2, b(%) = (1 +

p%)2.

Then

f%(x) =

p(b(%)� x)(x� a(%))

2⇡x [a(%),b(%)](x).

You may take it for granted that f% is a probability density when % � 1.

God

4. Show that, for any % > 0, f1/%(x) = f%(%x). Use this to show thatZ

Rf%(x) dx = % for % 2 (0, 1).

5. Let Y be a continuous random variable whose density is f1. Show thatpY has a

quarter-circular density:

fpY (x) =1

⇡

p4� x2

[0,2](x).

2

1.P is an orthogonal projection , i .

PEP P .- PTlU

If v. an eigenvector with eigenvalue X,

by a Pv.

= P2 , =P ( Py)= Play ) = XPy = X - by

-

- II.

Since v. is an eigenvector,

it is te , so a ;D .

1=0 or 1=1.

⇐ xx-D -o

ki : P, ⇒ is true for all ee f- GKP)Py -- e ⇒ ye dull (P) = 611 PT )

t

(Fundamental Theoremof Linear Algebra)

Smo Papi

⇒ Eigenspace of e is V'

Eigenspace of l is V

2. la) tfsij Ere lab = § lEijlaclnelcb'- §Siasjcfkcsebw

the only nonzero terms in the| sum have ⇐j and c -k

'

'

-

e Sia . Sjk . Seb= Sin Sia Seb = Sjr ( Eielab

i. Eij Ere = Sik Fie

.

(b) Tr ( XEij) - § EXEijlaa - ↳ ④abtEijlba=? Adab fibs ,a= ( Xlji .

⑥ Suppose X satisfies TrlXABI 'TRIXBA)for all matrices A

,B

.

That means,for all

"DHTrix Eij End = TRIX Ene Iii)

Tr Sheene) try"

- Sei Eng'd by ca)

Sju CX Ili Shi je by (b) .

Thus : Sin CX Ili = Sei Cxljn for all i. joke .

• Look at any j, k with jtk .

Take f- it.Thus

Sfi CX h ,z Si , cxlja i . lxljn --o

'

f j#k .' I

0 I

. Look at je k , and i=L .

i. Sjj LX Iii - Sii lxljji i[

i. all diagonal entries are equal , to somenumber X .

Thus x --= XI

.

3. Typeingueton : Gus . - z- ray

-

Z

i. Glxtig = -H - tvcatiyfhq

Se - ImGlxtiy) - § it Im ( IuetiyfT)Take lysine J toi. lying -¥ImGlxtiD=¥f¥nImHcxtiy

= Im flyingvcatiy= Im WHT)

If 10432 , x'- 4 >od

- : Bett EIR se Im 1=0.

If held 2 , ah- 4 soi.VH - f-147 =i4#

so Ima f- 14¥

i. fog > FYI -*InGlatt'D = ↳htt Dixie .

4. f. glass Vlk-ayDhIz

ACalf, bigot")

-i.fypuet-llx-azl.IM#Dcacyp.bNpllHf:::::i::i:ii: :c:::: : ::&

.

i. fight. VK.NO/ffofebkftf-T-Bcaipg,byxptD=rgt-tgVfx-alp7bIp-pT2€ Deals#

blpilpd.IE#m.o.uore" is:S :&,= fglpu) .

Now,if fat , Yf > I

,

and so

g. y .

fgkt-fyagfxt-f.gl#yn-:ffpiasdx=ffygh4uf.dyx...-asf?jyduW-fdn .- f - ffygluidn

5.

For f- I , all I - H -RT > ebut = ( HRT = 4 .

i. f. ex ) -- Vxn4Y Ice.ua)

= '%I4 Dee,41"" -

Now, if Y has density fy .- f, , then Go with

probability 1, and for 630,

PITY St) ' PH stye fluidic .

i. fry ftp.dqlplrystlzddg.tjflxldx= filly . adfltg

turn elementalTheorem of

= T4¥%,.pro#9kuhsJ= ¥r4tT I ↳ loft)( ble t'%4,⇒ -Lsts 2

but we know Pi 30So ftp.ltko if too .