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Chapter 7: Q4E (page 395)

Determine which of the matrices in Exercises 1–6 are symmetric.

4. \(\left( {\begin{aligned}{{}}0&8&3\\8&0&{ - 4}\\3&2&0\end{aligned}} \right)\)

Short Answer

Expert verified

The given matrix is not symmetric.

Step by step solution

01

Find the transpose

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

It is given that\(A = \left( {\begin{aligned}{{}}0&8&3\\8&0&{ - 4}\\3&2&0\end{aligned}} \right)\). Find the transpose of\(A\), as shown below:

\(\begin{aligned}{}{A^T} = \left( {\begin{aligned}{{}}0&8&3\\8&0&2\\3&{ - 4}&0\end{aligned}} \right)\\ \ne A\end{aligned}\)

02

Draw the conclusion

As \({A^T} \ne A\), so it can be concluded that \(A\) is not asymmetric matrix.

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Most popular questions from this chapter

Question: Let \({\bf{X}}\) denote a vector that varies over the columns of a \(p \times N\) matrix of observations, and let \(P\) be a \(p \times p\) orthogonal matrix. Show that the change of variable \({\bf{X}} = P{\bf{Y}}\) does not change the total variance of the data. (Hint: By Exercise 11, it suffices to show that \(tr\left( {{P^T}SP} \right) = tr\left( S \right)\). Use a property of the trace mentioned in Exercise 25 in Section 5.4.)

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

26.

  1. There are symmetric matrices that are not orthogonally diagonizable.
  2. b. If \(B = PD{P^T}\), where \({P^T} = {P^{ - {\bf{1}}}}\) and D is a diagonal matrix, then B is a symmetric matrix.
  3. c. An orthogonal matrix is orthogonally diagonizable.
  4. d. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

Suppose A is a symmetric \(n \times n\) matrix and B is any \(n \times m\) matrix. Show that \({B^T}AB\), \({B^T}B\), and \(B{B^T}\) are symmetric matrices.

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 17–24, \(A\) 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 \(m \times n\) “diagonal” 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.)
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