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Chapter 7: Q7E (page 395)

Determine which of the matrices in Exercises 7–12 are orthogonal. If orthogonal, find the inverse.

7. \(\left( {\begin{aligned}{{}{}}{.6}&{\,\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\)

Short Answer

Expert verified

\(P\) is an orthogonal matrix and\({P^{ - 1}} = \left( {\begin{aligned}{{}{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\).

Step by step solution

01

Find the characteristic equation

A matrix\(P\) with, \(n \times n\) dimension, is orthogonal if it satisfies the equation\({P^T}P = {I_n}\).

It is given that\(P = \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\). Find the matrix\({P^T}P\)as shown below:

\(\begin{aligned}{}{P^T}P &= \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&0\\0&1\end{aligned}} \right)\\ &= {I_2}\end{aligned}\)

02

Find the inverse

As\({P^T}P = {I_2}\), it can be concluded that\(P\)is an orthogonal matrix. So, the inverse of matrix\(P\)is\({P^T}\). Find\({P^T}\), as follows:

\(\begin{aligned}{}{P^{ - 1}} &= {P^T}\\ &= \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\end{aligned}\)

Thus, \(P\) is an orthogonal matrix and \({P^{ - 1}} = \left( {\begin{aligned}{{}}{.6}&{\,\,.8}\\{.8}&{ - .6}\end{aligned}} \right)\).

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Most popular questions from this chapter

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.

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

Let \(A = \left( {\begin{aligned}{{}}{\,\,\,2}&{ - 1}&{ - 1}\\{ - 1}&{\,\,\,2}&{ - 1}\\{ - 1}&{ - 1} &{\,\,\,2}\end{aligned}} \right)\),\({{\rm{v}}_1} = \left( {\begin{aligned}{{}}{ - 1}\\{\,\,\,0}\\{\,\,1}\end{aligned}} \right)\) and and\({{\rm{v}}_2} = \left( {\begin{aligned}{{}}{\,\,\,1}\\{\, - 1}\\{\,\,\,\,1}\end{aligned}} \right)\). Verify that\({{\rm{v}}_1}\), \({{\rm{v}}_2}\) an eigenvector of \(A\). Then orthogonally diagonalize \(A\).

Question: [M] The covariance matrix below was obtained from a Landsat 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 \({y_1}\)?

\[S = \left[ {\begin{array}{*{20}{c}}{29.64}&{18.38}&{5.00}\\{18.38}&{20.82}&{14.06}\\{5.00}&{14.06}&{29.21}\end{array}} \right]\]

In Exercises 25 and 26, mark each statement True or False. Justify each answer.

a. An\(n \times n\)matrix that is orthogonally diagonalizable must be symmetric.

b. If\({A^T} = A\)and if vectors\({\rm{u}}\)and\({\rm{v}}\)satisfy\(A{\rm{u}} = {\rm{3u}}\)and\(A{\rm{v}} = {\rm{3v}}\), then\({\rm{u}} \cdot {\rm{v}} = {\rm{0}}\).

c. An\(n \times n\)symmetric matrix has n distinct real eigenvalues.

d. For a nonzero \({\rm{v}}\) in \({\mathbb{R}^n}\) , the matrix \({\rm{v}}{{\rm{v}}^T}\) is called a projection matrix.

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