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

1. \(\left[ {\begin{aligned}{{}}3&{\,\,\,5}\\5&{ - 7}\end{aligned}} \right]\)

Short Answer

Expert verified

The given matrix is 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\(A = \left[ {\begin{aligned}{{}}3&{\,\,5}\\5&{ - 7}\end{aligned}} \right]\). Find the transpose of\(A\), as shown below:

\(\begin{aligned}{}{A^T} &= \left[ {\begin{aligned}{{}}3&{\,\,5}\\5&{ - 7}\end{aligned}} \right]\\ &= A\end{aligned}\)

02

Draw the conclusion

As \({A^T} = A\), so it can be concluded that \(A\) is a symmetric matrix.

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Most popular questions from this chapter

Question: 6. Let A be an \(n \times n\) symmetric matrix. Use Exercise 5 and an eigenvector basis for \({\mathbb{R}^n}\) to give a second proof of the decomposition in Exercise 4(b).

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

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

Compute the quadratic form \({{\bf{x}}^T}A{\bf{x}}\), when \(A = \left( {\begin{aligned}{{}}5&{\frac{1}{3}}\\{\frac{1}{3}}&1\end{aligned}} \right)\) and

a. \({\bf{x}} = \left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\)

b. \({\bf{x}} = \left( {\begin{aligned}{{}}6\\1\end{aligned}} \right)\)

c. \({\bf{x}} = \left( {\begin{aligned}{{}}1\\3\end{aligned}} \right)\)

Construct a spectral decomposition of A from Example 2.

Question: In Exercises 15 and 16, construct the pseudo-inverse of \(A\). Begin by using a matrix program to produce the SVD of \(A\), or, if that is not available, begin with an orthogonal diagonalization of \({A^T}A\). Use the pseudo-inverse to solve \(A{\rm{x}} = {\rm{b}}\), for \({\rm{b}} = \left( {6, - 1, - 4,6} \right)\) and let \(\mathop {\rm{x}}\limits^\^ \)be the solution. Make a calculation to verify that \(\mathop {\rm{x}}\limits^\^ \) is in Row \(A\). Find a nonzero vector \({\rm{u}}\) in Nul\(A\), and verify that \(\left\| {\mathop {\rm{x}}\limits^\^ } \right\| < \left\| {\mathop {\rm{x}}\limits^\^ + {\rm{u}}} \right\|\), which must be true by Exercise 13(c).

15. \(A = \left[ {\begin{array}{*{20}{c}}{ - 3}&{ - 3}&{ - 6}&6&{\,\,1}\\{ - 1}&{ - 1}&{ - 1}&1&{ - 2}\\{\,\,\,0}&{\,\,0}&{ - 1}&1&{ - 1}\\{\,\,\,0}&{\,\,0}&{ - 1}&1&{ - 1}\end{array}} \right]\)

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