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Chapter 7: Q12E (page 395)

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

12. \(P = \left( {\begin{aligned}{{}}{.5}&{.5}&{ - .5}&{ - .5}\\{.5}&{.5}&{.5}&{.5}\\{.5}&{ - .5}&{ - .5}&{.5}\\{.5}&{ - .5}&{.5}&{ - .5}\end{aligned}} \right)\)

Short Answer

Expert verified

The given matrix is an orthogonal matrix, and\({P^{ - 1}} = \left( {\begin{aligned}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\end{aligned}} \right)\).

Step by step solution

01

Find the characteristic equation

A matrix\(P\) with, \(n \times n\) dimension, is orthogonal if it satisfies the equation\({P^T}P = {I_n}\)and its inverse is given as \({P^{ - 1}} = {P^T}\).

It is given that\(P = \left( {\begin{aligned}{{}}{.5}&{.5}&{ - .5}&{ - .5}\\{.5}&{.5}&{.5}&{.5}\\{.5}&{ - .5}&{ - .5}&{.5}\\{.5}&{ - .5}&{.5}&{ - .5}\end{aligned}} \right)\).

Find the matrix\({P^T}P\)as shown below:

\(\begin{aligned}{}{P^T}P &= \left( {\begin{aligned}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\end{aligned}} \right)\left( {\begin{aligned}{{}}{.5}&{.5}&{ - .5}&{ - .5}\\{.5}&{.5}&{.5}&{.5}\\{.5}&{ - .5}&{ - .5}&{.5}\\{.5}&{ - .5}&{.5}&{ - .5}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{aligned}} \right)\\ &= {I_4}\end{aligned}\)

02

Find the inverse

As\({P^T}P = {I_4}\), it can be concluded that\(P\)is an orthogonal matrix. So, the inverse of matrix\(P\)is\({P^T}\).

Find\({P^T}\)as follows:

\(\begin{aligned}{}{P^{ - 1}}& = {P^T}\\ &= \left( {\begin{aligned}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\end{aligned}} \right)\end{aligned}\)

Thus, the given matrix is an orthogonal matrix, and\({P^{ - 1}} = \left( {\begin{aligned}{{}}{.5}&{.5}&{\,.5}&{\,.5}\\{.5}&{.5}&{ - .5}&{ - .5}\\{ - .5}&{.5}&{ - .5}&{\,\,.5}\\{ - .5}&{.5}&{\,\,\,.5}&{ - .5}\end{aligned}} \right)\).

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Most popular questions from this chapter

In Exercises 17鈥�24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

23. Let \(U = \left( {{u_1}...{u_m}} \right)\) and \(V = \left( {{v_1}...{v_n}} \right)\) where the \({{\bf{u}}_i}\) and \({{\bf{v}}_i}\) are in Theorem 10. Show that \(A = {\sigma _1}{u_1}v_1^T + {\sigma _2}{u_2}v_2^T + ... + {\sigma _r}{u_r}v_r^T\).

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.

Question:Find the principal components of the data for Exercise 1.

In Exercises 17鈥�24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

21. Justify the statement in Example 2 that the second singular value of a matrix \(A\) is the maximum of \(\left\| {A{\bf{x}}} \right\|\) as \({\bf{x}}\) varies over all unit vectors orthogonal to \({{\bf{v}}_{\bf{1}}}\), with \({{\bf{v}}_{\bf{1}}}\) a right singular vector corresponding to the first singular value of \(A\). (Hint: Use Theorem 7 in Section 7.3.)

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17鈥�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.

16. \(\left( {\begin{aligned}{{}}{\,6}&{ - 2}\\{ - 2}&{\,\,\,9}\end{aligned}} \right)\)
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