Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered, Verified

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




vii

,Contents


1 Linear Algebra and Optimization: An Introduction
u z u z u z u z u z 1


2 Linear Transformations and Linear Systems
u z u z u z u z 17


3 Diagonalizable Matrices and Eigenvectors u z u z u z 35


4 Optimization Basics: A Machine Learning View uz uz uz uz uz 47


5 Optimization Challenges and Advanced Solutions u z u z u z u z 57


6 Lagrangian Relaxation and Duality u z u z u z 63


7 Singular Value Decomposition u z u z 71


8 Matrix Factorization u z 81


9 The Linear Algebra of Similarity
u z u z u z u z 89


10 The Linear Algebra of Graphs
u z u z u z u z 95


11 Optimization in Computational Graphs u z u z u z 101




viii

,Chapter 1 u z




Linear Algebra and Optimization: An Introduction
uz uz uz uz uz




1. For any two vectors x and y, which are each of length a, show that (i) x
u z u z u z u z u z u z u z u z u z u z u z u z u z u z u z u z uz




− y is orthogonal to x+y, and (ii) the dot product of x− 3y and x+3y is negativ
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e.
(i) The first is simply·x −x · y y using the distributive property of matrix multi
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u z u z uz uz uz uz uz uz uz




plication. The dot product of a vector with itself is its squared length. Since b uz uz uz uz uz uz uz uz uz uz uz uz uz uz




oth vectors are of the same length, it follows that the result is 0. (ii) In the seco
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nd case, one can use a similar argument to show that the result is a2 − 9a2, whi
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ch is negative. uz uz




2. Consider a situation in which you have three matrices A, B, and C, of sizes 10 u z u z u z u z u z u z u z u z u z u z u z u z u z u z u z zu




×2, 2×10, and 10×10, respectively.
zu uz zu zu uz u z zu zu u z




(a) Suppose you had to compute the matrix product ABC. From an efficiency pe uz uz uz uz uz uz uz uz uz uz uz uz




r-
spective,would itcomputationallymake moresensetocompute(AB)C orwoul
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d it make more sense to compute A(BC)?
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(b) If you had to compute the matrix product CAB, would it make more sense to c
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ompute (CA)B or C(AB)? u z u z u z




The main point is to keep the size of the intermediate matrix as small as po
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ssible inorder to reduce both computational and spacerequirements. In t u z uz uz uz uz uz uz uz uz uz uz




he case of ABC, it makes sense to compute BC first. In the case of CAB it ma
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kes sense to compute CA first.This type of associativity property isused fr
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equently inmachine learningin order to reduce computational requireme uz uz uz uz uz uz uz uz uz




nts.
3. Show that if a matrix A satisfies —
A = u z u z u z u z u z u z u z u z




AT, then all the diagonal elements of th uz u z u z u z u z u z u z u z




e matrix are 0.
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NotethatA+AT=0.However,thismatrix alsocontains twice the diagonal
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elements of A on its diagonal. Therefore, the diagonal elements of A must uz uz uz uz uz uz uz uz uz uz uz uz uz




be 0. uz




4. Show that if we have a matrix satisfying A—= uz uz uz uz uz uz uz uz uz




1

, AT, then for any column vector x, we
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have x Ax = 0. u z
T
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Note that the transpose of the scalar xTAx remains unchanged. Therefore,
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we haveu z




xTAx = (xTAx)T = xTATx = −xTAx. Therefore, we have 2xTAx = 0.
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2