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SOLUTION MANUAL Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal. Chapters 1 – 11 $17.99   Add to cart

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SOLUTION MANUAL Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal. Chapters 1 – 11

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SOLUTION MANUAL Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal. Chapters 1 – 11

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  • September 17, 2024
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SOLUTION MANUAL
Linear Algebra and Optimization for Machine
Learning
1st Edition by Charu Aggarwal. Chapters 1 – 11




vii

,Contents


1 Linearm Algebram andm Optimization:m Anm Introduction 1


2 Linearm Transformations m andm Linearm Systems 17


3 Diagonalizablem Matricesm andm Eigenvectors 35


4 OptimizationmBasics:mAmMachinemLearningmView 47


5 Optimizationm Challengesm andm Advancedm Solutions 57


6 Lagrangianm Relaxationm andm Duality 63


7 Singularm Valuem Decomposition 71


8 Matrixm Factorization 81


9 Them Linearm Algebram ofm Similarity 89


10 Them Linearm Algebram ofm Graphs 95


11 Optimizationm inm Computationalm Graphs 101




viii

,Chapterm 1

LinearmAlgebramandmOptimization:mAnmIntroduction




1. Form anym twom vectorsm xm andm y,m whichm arem eachm ofm lengthm a,m showm thatm (i
)m xm−mym ismorthogonalmtomxm+my,m andm(ii)m themdotmproductmofmxm−m3ym andmxm
+m3ym ism negative.
(i)mThemfirstmismsimply
·m −mmx·m xm ym ymusingmthemdistributivempropertymofmmatri
xmmultiplication.mThemdotmproductmofmamvectormwithmitselfmismitsmsquaredm
length.mSincembothmvectorsmaremofmthemsamemlength,mitmfollowsmthatmthemr
esultmism0.m(ii)mInmthemsecondmcase,monemcanmusemamsimilarmargumentmtomsh
owmthatmthemresultmisma2m−m9a2,mwhichmismnegative.
2. Considerm am situationm inm whichm youm havem threem matricesm A,m B,m andm C,m ofm si
zesm 10m×m2,m2m×m10,mandm10m×m10,mrespectively.
(a) SupposemyoumhadmtomcomputemthemmatrixmproductmABC.mFrommanmefficie
ncymper-
mspective,mwouldmitmcomputationallymmakemmoremsensemtomcomputem(AB)Cm

ormwouldmitmmakemmoremsensemtomcomputemA(BC)?
(b) IfmyoumhadmtomcomputemthemmatrixmproductmCAB,mwouldmitmmakemmorems
ensemtomcomputem (CA)Bm orm C(AB)?
Themmainmpointmismtomkeepmthemsizemofmthemintermediatemmatrixmasms
mallmasmpossiblem inmordermtomreducembothmcomputationalmandmspacemr
equirements.mInmthemcasemofmABC,mitmmakesmsensemtomcomputemBCmfirs
t.mInmthemcasemofmCABmitmmakesmsensemtomcomputemCAmfirst.mThismtype
mofmassociativitympropertymismused mfrequentlyminmmachinemlearningminm

ordermtomreducemcomputationalmrequirements.
3. Showm thatm ifm am matrixm Am satisfies—m Am =
ATm,m thenm allm them diagonalm elementsm o
fm themmatrixmarem0.
NotemthatmAm+mATm=m0.mHowever,mthismmatrixmalsomcontainsmtwicemthe
mdiagonalmelementsmofm Amonmitsm diagonal. mTherefore,mthemdiagonal mele

mentsmofmAmmustmbem0.
4. Showmthatmifmwemhavemammatrixmsatisfying
— mAm=
1

, ATm,mthenmformanymcolumnmvectormx
,mwemhavem x mAxm=m0.
T


Notem thatm them transposem ofm them scalarm xTmAxm remainsm unchanged.m Therefore,
m wem have




xTmAxm=m(xTmAx)Tm =mxTmATmxm=m−xTmAx.m Therefore,m wem havem 2xTmAxm=
m0.




2

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