Chapter 7: Q1E (page 395)
Determine which of the matrices in Exercises 1–6 are symmetric.
1. \(\left[ {\begin{aligned}{{}}3&{\,\,\,5}\\5&{ - 7}\end{aligned}} \right]\)
Short Answer
The given matrix is symmetric.
Chapter 7: Q1E (page 395)
Determine which of the matrices in Exercises 1–6 are symmetric.
1. \(\left[ {\begin{aligned}{{}}3&{\,\,\,5}\\5&{ - 7}\end{aligned}} \right]\)
The given matrix is symmetric.
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Get started for freeQuestion: 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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