faemncrljrl/teaching/cse599isp... · 2021. 6. 2. · some homework format casestudy...
TRANSCRIPT
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