Some homeworkFormat Case study

classical15calur phenomenon 4 PSD analog

nxn real matrices Mn M.nlIRA CMn symmetric A ATA is positiedetinitee if A is symmetricand

Lx Ax 70 AXERn

PSI if 4 Ax zo earnit turn

Real symmetric matrices can be diagonalizedA UDUT UU'T Utu I

D diag X Xz H D diagonalAni XiUi U ly lyn

A symmetric AisPSDC 4 hu r.hn

MtClR fAEMnCRl A is PSD

pclosed convex A IBE tht IR At CB C Mutant

cone H C o

LXI toB x Lx Ax thx Bx

A e EX unlit MutCR cone uut weird

µ psp matrix a i sa bi

A Mucci is Hermitian if A A

Lx Ax L Atx x symmetricSelf adjointI

Ea ieeumin.inm

AEMnCQisPsDifLx AxznZ0V xECn

EemiijanuD digs W

L XIE Ryu u u I

unitary

Quantum informationtheoryclassical quantum

probabilitydistribution density matrix AEMucaA PSD Tr CA L

pEI

diasc

bucceq A o A is pos defLoewver order AZ 0 A is PSD

A E B A B to

Cx Ax z Lx BxFX ERM

Grammat XE Mun IR Mxn real matrix

A X E Muck

X XnElRm Aij sxi xp X Y lyn

Lu Au Lie XTXu Xu Xu HXuHEo

A PSD A hiuiy.tl i2oCui3ovthonomd

FA TxiuiuT FATA XiniuitA

A FA FA Aijcxi.xjsxt.xm.snra tin

M

Given x Xn E Rm then F XT In EIRsit Lili I Lxi xp Hii

ExI If 4 Lz hn O and AE Min is defby

Aij xi

Then A is PSD

g f e tkitmdt tfi.fjhq o.my

IIticitiaCf

t p STE ft n3

Claim AzoPETO's

As t E Ys Tt Ys IT XiIES

Xi iidfo13r.VETesuxiJ plsMp xi 1y p

Consider X Xz Xn E Rm There exist

Yuya syn EIR s.tolyHz lynHz 1

C miulllxi xjlh.IT lyi yjHzECyninCHXi xj 1 tiis

x Cc _ie

r

f gj ggYjyiERfexplicxj.gs gig gnStandard Gaussian

fjx Cid NCO 1

T Eight to

If explisxi xj.gs 1TLxi xj.gs

to llxi xjllz.gs

4 jig KX.gl zgs

E expCillxi xjHg 12

2 El cos Aki xjlh.gl H

A BE OAB symmetric AB BA

AB sinned diag

Hadamard product Ao B ij Ai Bisaka Schur product

T.IR 7112N

Facet AIB EO then linearmap

A cMn

eA T B TAtBA qaiuiu.itB bivjVjt Lej Tale D Aij

AIB 20 then TAoTB

caTI iosgigs

Lll

ngA B Zo L not nea psd

Fa HA40and A thenBE0

A diag a.tn an ti

taCAB BA B off diffB tCAB A off 2,1 E 0

ABI matrixsort

CLai B A E opise Foe

AZ BZ a b atb

X2 Y s X Y LXtY t YIN YD

F Y2 and XtY o X Y to

AIB Fates o XZY

X FA TAE BrY FB

I 53

A I 3 B 13It true that At B AZ EB

hE

t p SEEn3

Claim AZ 0

Ast EF Eea

Ia

HIs Xi

11

he Said Fid can

Let CR µ be a pub space LF S IR

Lef f the family of r.us or din

Jfscw f Cw dµCw

yse.IE

If g a Jo fat sit dt

S o

EpsA

AijB

A B ij BIff

AiiBij iijj

IIE'Yumn A c M

Le ej A B ek eeD ijke

IRM Rn 1km

E em e yen e Ej

Vij Lu ei ej'sVij i Ej iEEm

jEEnTfA0xB7ci.jli's't Aim Bjj