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Q38E

Expert-verified
Found in: Page 393

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

If A is any symmetric 2x2 matrix with eigenvalues -2 and 3, and $\stackrel{\mathbf{\to }}{\mathbf{u}}$ is a unit vector ${{\mathbf{ℝ}}}^{{\mathbf{2}}}{\mathbf{,}}{{\mathbf{ℝ}}}^{{\mathbf{2}}}$ , what are the possible values of the dot product $\stackrel{\mathbf{\to }}{\mathbf{u}}{\mathbf{×}}{\mathbf{A}}\stackrel{\mathbf{\to }}{\mathbf{u}}$ ? Illustrate your answer, in terms of the unit circle and its image A .

The possible values of the dot product $-2\le \stackrel{\to }{u}.A\stackrel{\to }{u}\le 3,$ where a unit vector is

See the step by step solution

Step 1: Symmetric matrix:

In linear algebra, a symmetric matrix is a square matrix that stays unchanged when its transpose is calculated. In that instance, a symmetric matrix is one whose transpose equals the matrix itself.

Step 2: Find the possible values of the dot product u→×Au→ :

Given that, A is any symmetric 2x2 matrix with eigenvalues -2 and 3, and is a unit vector ${\mathrm{ℝ}}^{2}$ . From the spectral theorem, we know that there exists an orthonormal eigen basis ${\stackrel{\to }{v}}_{1},{\stackrel{\to }{v}}_{2}$ for T , with associated real eigenvalues ${\lambda }_{1}=3$ and ${\lambda }_{2}=-2$ (Arrange things so that $\left|{\lambda }_{1}\right|\ge \left|{\lambda }_{2}\right|$). Now consider the unit vector $\stackrel{\to }{u}$ is represented below:

$\stackrel{\to }{u}={c}_{1}{\stackrel{\to }{v}}_{1}+{c}_{2}{\stackrel{\to }{v}}_{2}\phantom{\rule{0ex}{0ex}}⇒A\stackrel{\to }{u}={\lambda }_{1}{c}_{1}{\stackrel{\to }{v}}_{1}+{\lambda }_{2}{c}_{2}{\stackrel{\to }{v}}_{2}\phantom{\rule{0ex}{0ex}}⇒A\stackrel{\to }{u}=3{c}_{1}{\stackrel{\to }{v}}_{1}+2{c}_{2}{\stackrel{\to }{v}}_{2}\phantom{\rule{0ex}{0ex}}$

Now evaluate $\stackrel{\to }{u}.A\stackrel{\to }{u}$ as follows:

role="math" localid="1659612634271" $\stackrel{\to }{u}.A\stackrel{\to }{u}=\left({c}_{1}{\stackrel{\to }{v}}_{1}+{c}_{2}{\stackrel{\to }{v}}_{2}\right).\left(3{c}_{1}{\stackrel{\to }{v}}_{1}-2{c}_{2}{\stackrel{\to }{v}}_{2}\right)\phantom{\rule{0ex}{0ex}}⇒\stackrel{\to }{u}.A\stackrel{\to }{u}=3{c}_{1}^{2}-2{c}_{2}^{2}\left(1\right)$

Since

$3{c}_{1}^{2}-2{c}_{2}^{2}\le \left(3{c}_{1}^{2}+3{c}_{2}^{2}=3\right)\left(2\right)$

and

$\left(-2{c}_{2}^{2}-2{c}_{2}^{2}=-2\right)\le 3{c}_{1}^{2}-2{c}_{2}^{2}\left(3\right)$

From (1), (2) and (3) we can imply that the possible values of the dot product $\stackrel{\to }{u}.A\stackrel{\to }{u}$ is as represented below:

$-2\le \stackrel{\to }{u}.A\stackrel{\to }{u}\le 3$

Step 3: The plot of the unit circle and its image:

The orthonormal Eigen values are here ${\lambda }_{1}=3$ is positive and ${\lambda }_{2}=-2$ is negative. The plot of the unit circle and its image under A is represented below.