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Q7.5-5E

Expert-verified

Linear Algebra and its Applications

Found in: Page 395

Book edition 5th

Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald

Pages 483 pages

ISBN 978-03219822384

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Short Answer

Question: [M] A Landsat image with three spectral components was made of Homestead Air Force Base in Florida (after the base was hit by Hurricane Andrew in 1992). The covariance matrix of the data is shown below. Find the first principal component of the data, and compute the percentage of the total variance that is contained in this component.

**\[S = \left[ {\begin{array}{*{20}{c}}{164.12}&{32.73}&{81.04}\\{32.73}&{539.44}&{249.13}\\{81.04}&{246.13}&{189.11}\end{array}} \right]\]**

The required percentage is \(75.8956\% \).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Mean Deviation form and Covariance Matrix

The Mean Deviation form of any \(p \times N\) is given by:

\(B = \left[ {\begin{array}{*{20}{c}}{{{\hat X}_1}}&{{{\hat X}_2}}&{........}&{{{\hat X}_N}}\end{array}} \right]\)

Whose \(p \times p\) covariance matrix is:

\(S = \frac{1}{{N - 1}}B{B^T}\)

Step 2: The Variance

From question, the covariance matrix and the maximum eigenvalue we haveis:

\(\begin{array}{l}S = \left[ {\begin{array}{*{20}{c}}{164.12}&{32.73}&{81.04}\\{32.73}&{539.44}&{249.13}\\{81.04}&{246.13}&{189.11}\end{array}} \right]\\{\lambda _1} = 677.497\end{array}\)

The respective unit vector is:

\({u_1} = \left[ {\begin{array}{*{20}{c}}{0.129554}\\{0.874423}\\{0.467547}\end{array}} \right]\)

Now, the percentage of change in variance can be obtained as:

\[\begin{array}{c}\Delta = \frac{{{\lambda _1}}}{{tr\left( S \right)}} \times 100\\ = \frac{{677.4978}}{{164.12 + 539.44 + 189.11}} \times 100\\ = 75.8956\% \end{array}\]

Hence, this is the required answer.

Most popular questions for Math Textbooks

Question: Mark Each statement True or False. Justify each answer. In each part, A represents an \(n \times n\) matrix.

If A is orthogonally diagonizable, then A is symmetric.
If A is an orthogonal matrix, then A is symmetric.
If A is an orthogonal matrix, then \(\left\| {A{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\) for all x in \({\mathbb{R}^n}\).
The principal axes of a quadratic from \({{\bf{x}}^T}A{\bf{x}}\) can be the columns of any matrix P that diagonalizes A.
If P is an \(n \times n\) matrix with orthogonal columns, then \({P^T} = {P^{ - {\bf{1}}}}\).
If every coefficient in a quadratic form is positive, then the quadratic form is positive definite.
If \({{\bf{x}}^T}A{\bf{x}} > {\bf{0}}\) for some x, then the quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is positive definite.
By a suitable change of variable, any quadratic form can be changed into one with no cross-product term.
The largest value of a quadratic form \({{\bf{x}}^T}A{\bf{x}}\), for \(\left\| {\bf{x}} \right\| = {\bf{1}}\) is the largest entery on the diagonal A.
The maximum value of a positive definite quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is the greatest eigenvalue of A.
A positive definite quadratic form can be changed into a negative definite form by a suitable change of variable \({\bf{x}} = P{\bf{u}}\), for some orthogonal matrix P.
An indefinite quadratic form is one whose eigenvalues are not definite.
If P is an \(n \times n\) orthogonal matrix, then the change of variable \({\bf{x}} = P{\bf{u}}\) transforms \({{\bf{x}}^T}A{\bf{x}}\) into a quadratic form whose matrix is \({P^{ - {\bf{1}}}}AP\).
If U is \(m \times n\) with orthogonal columns, then \(U{U^T}{\bf{x}}\) is the orthogonal projection of x onto ColU.
If B is \(m \times n\) and x is a unit vector in \({\mathbb{R}^n}\), then \(\left\| {B{\bf{x}}} \right\| \le {\sigma _{\bf{1}}}\), where \({\sigma _{\bf{1}}}\) is the first singular value of B.
A singular value decomposition of an \(m \times n\) matrix B can be written as \(B = P\Sigma Q\), where P is an \(m \times n\) orthogonal matrix and \(\Sigma \) is an \(m \times n\) diagonal matrix.
If A is \(n \times n\), then A and \({A^T}A\) have the same singular values.

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17–22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

18. \(\left( {\begin{aligned}{{}}1&{ - 6}&4\\{ - 6}&2&{ - 2}\\4&{ - 2}&{ - 3}\end{aligned}} \right)\)

Let u be a unit vector in \({\mathbb{R}^n}\), and let \(B = {\bf{u}}{{\bf{u}}^T}\).

Given any x in \({\mathbb{R}^n}\), compute Bx and show that Bx is the orthogonal projection of x onto u, as described in Section 6.2.
Show that B is a symmetric matrix and \({B^{\bf{2}}} = B\).
Show that u is an eigenvector of B. What is the corresponding eigenvalue?

Find the matrix of the quadratic form. Assume x is in \({\mathbb{R}^2}\).

a. \(5x_1^2 + 16{x_1}{x_2} - 5x_2^2\)

b. \(2{x_1}{x_2}\)

Question: Repeat Exercise 7 for the data in Exercise 2.

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