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Chapter 7: Q3E (page 395)

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

3. \(\left( {\begin{aligned}{{}}2&{\,\,3}\\{\bf{2}}&4\end{aligned}} \right)\)

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\(A = \left( {\begin{aligned}{{}}2&{\,\,3}\\2&4\end{aligned}} \right)\). Find the transpose of\(A\), as shown below:

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

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

In Exercises 3-6, find (a) the maximum value of \(Q\left( {\rm{x}} \right)\) subject to the constraint \({{\rm{x}}^T}{\rm{x}} = 1\), (b) a unit vector \({\rm{u}}\) where this maximum is attained, and (c) the maximum of \(Q\left( {\rm{x}} \right)\) subject to the constraints \({{\rm{x}}^T}{\rm{x}} = 1{\rm{ and }}{{\rm{x}}^T}{\rm{u}} = 0\).

4. \(Q\left( x \right) = 3x_1^2 + 3x_2^2 + 5x_3^2 + 6x_1^{}x_2^{} + 2x_1^{}x_3^{} + 2x_2^{}x_3^{}\).

Question: If A is \(m \times n\), then the matrix \(G = {A^T}A\) is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 9 and 10).

9. Show that the Gram matrix of any matrix A is positive semidefinite, with the same rank as A. (See the Exercises in Section 6.5.)

Question: In Exercises 1 and 2, convert the matrix of observations to mean deviation form, and construct the sample covariance matrix.

\(1.\,\,\left( {\begin{array}{*{20}{c}}{19}&{22}&6&3&2&{20}\\{12}&6&9&{15}&{13}&5\end{array}} \right)\)

Question:(M) Compute the singular values of the \({\bf{5 \times 5}}\) matrix in Exercise 10 in Section 2.3, and compute the condition number \(\frac{{{\sigma _1}}}{{{\sigma _{\bf{5}}}}}\).

In Exercises 1 and 2,find the change of variable \({\rm{x}} = P{\rm{y}}\) that transforms the quadratic form \({{\rm{x}}^T}A{\rm{x}}\) into \({{\rm{y}}^T}D{\rm{y}}\) as shown.

2. \(3x_1^2 + 3x_2^2 + 5x_3^2 + 6x_1^{}x_2^{} + 2x_1^{}x_3^{} + 2x_2^{}x_3^{} = 7y_1^2 + 4y_2^2\).
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