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$[\mathbf{M}]$ The covariance matrix below was obtained from a Land-sat image of the Columbia River in Washington, using data from three spectral bands. Let $x_{1}, x_{2}, x_{3}$ denote the spectral components of each pixel in the image. Find a new variable of the form $y_{1}=c_{1} x_{1}+c_{2} x_{2}+c_{3} x_{3}$ that has maximum possible variance, subject to the constraint that $c_{1}^{2}+c_{2}^{2}+c_{3}^{2}=1$ What percentage of the total variance in the data is explained by $y_{1} ?$$$S=\left[\begin{array}{ccc}{29.64} & {18.38} & {5.00} \\ {18.38} & {20.82} & {14.06} \\ {5.00} & {14.06} & {29.21}\end{array}\right]$$

64.8872$\%$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 5

Applications to Image Processing and Statistics

Introduction to Matrices

Missouri State University

Harvey Mudd College

Baylor University

Lectures

01:32

In mathematics, the absolu…

01:11

01:09

[M] A Landsat image with t…

01:08

The sample covariance matr…

01:13

Let $x_{1}, x_{2}$ denote …

02:53

Given multivariate data $\…

01:14

Suppose three tests are ad…

07:11

(a) Use the properties of …

02:50

Let $\mathbf{X}$ denote a …

08:35

(a) Compute the covariance…

17:13

Correlated bit noise. Let …

01:26

In Exercises 1 and $2,$ co…

this question. We have to find a new variable off the form. Why? One equal two C one x one plus seats. Who? X two, plus sit. Very ex Terry, that ax maximum possible very on subjects to the fact that C one squared. Plus it's a squared. Plus C three squared equals one to do this. No, What we do is that we find the large ists I give value office. We're s is the co variance matrix. Now what we do to find the again Vado, we do determinants off swings in 9.64 minus lambda. It's in points three aides five. It's important. Three seeds swing See points It's two minus Lunda 14.6 five 14.6 Swenton nine points to one minus Lambda. They said the District Court zero. When you solve these you have That's your lambda. One is 51.6957 You get your second. I give violets Will be 24.5351 on your thought wants to be 3.4392 So we see that the largest I give value is this. Now what do we do next, we find it in the eye again. Vento. Next, find that I give Victor corresponding Sue Lunda one. So how do you find it? I give it. So you plug in London one into these. And there you were, reduced that. And then you multiply that by X one x to exterior and sets the course zero. When you do that, you're gonna have That's the I give it so for lumber want equality. So 51.6957 is one point Sue. 02941 points. One several. 147 And they want I now recall that we want we won't see one squared plus cities Scripless citrus Quick was the one. So we normalized this idea Inventor so normalize by dividing by dividing but squared off one point Susie Rosa's knife or squared plus one points one several. +147 squared plus one squared. We normalize. You're gonna get 0.6155250 point 599 Ford Sue four on the 0.5116 It's three. So this will be our c one sea to silvery. So does the choice for the new variable new variable. Why want these? Why want equal soon? C one, which is 0.6155 to 5 more. Display my ex one plus it's Who? Most clever X to them, plus sit cherie 1116 It's three x three. So we saw the first part of the question. The second parts access. What percentage of interference in the data is explained by why Juan to get days so it gets the percentage off to Tove. Erian's explained. By y one. We do Lunda worn, which is again value the largest I given. Do they divide these by the trees off s wasa Truth is a song off the diagonal elements. So this refused to 1.6957 divided by swims in 9.64 plus 20 points. It's soon lost trains in nine points to want Where do the second half points? 64 It's 87 soon, so this implies what more supplies by. On judge that implies are 64 points. It's it's seven super scents variance. That's ah is explained, but why one

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In mathematics, the absolute value or modulus |x| of a real number x is its …

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