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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 41EQ

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The covariance matrix of an image with three spectral components is shown below. Let x₁, x₂, and x3 denote the spectral components of each pixel in the image. Find
a new variable of the form y₁ = C₁X₁ + C₂X₂ + C3x3 that has maximum possible variance, subject to the constraint that c3 + c3 + c3 = 1. What percentage of the total
variance in the data is explained by y₁?
27.22 19.55 6.00
S= 19.55 21.88 13.36
6.00 13.36 25.32
(...)
Which of the following could be the new variable y₁?
O A. Y₁ = -0.56x₁ +0.06x2 -0.83x3
O C.
y₁ = 0.62x₁ +0.62x2 +0.62x3
O B. y₁ = -0.54x₁ -0.78x2 +0.32x3
ⒸD. y₁=0.54x₁1 -0.78x2 + 0.32x3
O F. y₁ = 0.62x₁ +0.63x2 +0.47x3
y₁ = 0.62x₁ +0.63x2 -0.47x3
O E. y₁=0.54x₁1 -0.78x2 -0.32x3
O G. y₁ = 0.56x₁ +0.06x2 -0.83x3
O H.

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