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2. Problem: PCA with some analytical skills related questions. Consider the general PCA problem: Given a matrix of data, X of size p × N (p < N), we want to find the "best" approximation (in the average norm square sense) of xi≈µ+Vqλi, where V, is a unitary px q (q< p) matrix, and λ; a q x 1 vector, where x is the ith column of X. Denote the eigenvalues of XXT by V1, V2, ... ,Vp, in descending order. (a) Explain what happens to the PCA analysis, if xk = V₁, for some k in {1,2,,N}. In other words, what happens to the approximation of X? Note that v₁ is the eigenvector of XXT that corresponds to its maximum eigenvalue. (b) Describe what happens to PCA, if eigenvalues of XX v₁ = 0, for ≥ 91, with and q₁ > q. In other words, is the accuracy of the PCA better? 91

2. Problem: PCA with some analytical skills related questions. Consider the general PCA problem: Given a matrix of data, X of size p × N (p < N), we want to find the "best" approximation (in the average norm square sense) of xi≈µ+Vqλi, where V, is a unitary px q (q< p) matrix, and λ; a q x 1 vector, where x is the ith column of X. Denote the eigenvalues of XXT by V1, V2, ... ,Vp, in descending order. (a) Explain what happens to the PCA analysis, if xk = V₁, for some k in {1,2,,N}. In other words, what happens to the approximation of X? Note that v₁ is the eigenvector of XXT that corresponds to its maximum eigenvalue. (b) Describe what happens to PCA, if eigenvalues of XX v₁ = 0, for ≥ 91, with and q₁ > q. In other words, is the accuracy of the PCA better? 91

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 3AEXP
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2. Problem: PCA with some analytical skills related questions. Consider the general PCA problem: Given a matrix of data, X of size p × N (p < N), we want to find the "best" approximation (in the average norm square sense) of xi≈µ+Vqλi, where V, is a unitary px q (q< p) matrix, and λ; a q x 1 vector, where x is the ith column of X. Denote the eigenvalues of XXT by V1, V2, ... ,Vp, in descending order. (a) Explain what happens to the PCA analysis, if xk = V₁, for some k in {1,2,,N}. In other words, what happens to the approximation of X? Note that v₁ is the eigenvector of XXT that corresponds to its maximum eigenvalue. (b) Describe what happens to PCA, if eigenvalues of XX v₁ = 0, for ≥ 91, with and q₁ > q. In other words, is the accuracy of the PCA better? 91
Transcribed Image Text:2. Problem: PCA with some analytical skills related questions. Consider the general PCA problem: Given a matrix of data, X of size p × N (p < N), we want to find the "best" approximation (in the average norm square sense) of xi≈µ+Vqλi, where V, is a unitary px q (q< p) matrix, and λ; a q x 1 vector, where x is the ith column of X. Denote the eigenvalues of XXT by V1, V2, ... ,Vp, in descending order. (a) Explain what happens to the PCA analysis, if xk = V₁, for some k in {1,2,,N}. In other words, what happens to the approximation of X? Note that v₁ is the eigenvector of XXT that corresponds to its maximum eigenvalue. (b) Describe what happens to PCA, if eigenvalues of XX v₁ = 0, for ≥ 91, with and q₁ > q. In other words, is the accuracy of the PCA better? 91
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