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Q38E

Expert-verifiedFound in: Page 393

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 $\overrightarrow{\mathbf{u}}$** **is a unit vector ${{\mathbf{\mathbb{R}}}}^{{\mathbf{2}}}{\mathbf{,}}{{\mathbf{\mathbb{R}}}}^{{\mathbf{2}}}$** **, what are the possible values of the dot product $\overrightarrow{\mathbf{u}}{\mathbf{\times}}{\mathbf{A}}\overrightarrow{\mathbf{u}}$ ****? Illustrate your answer, in terms of the unit circle and its image A ****.**

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

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.

Given that, A is any symmetric 2x2 matrix with eigenvalues -2 and 3, and is a unit vector ${\mathrm{\mathbb{R}}}^{2}$ . From the spectral theorem, we know that there exists an orthonormal eigen basis ${\overrightarrow{v}}_{1},{\overrightarrow{v}}_{2}$ 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 $\overrightarrow{u}$ is represented below:

$\overrightarrow{u}={c}_{1}{\overrightarrow{v}}_{1}+{c}_{2}{\overrightarrow{v}}_{2}\phantom{\rule{0ex}{0ex}}\Rightarrow A\overrightarrow{u}={\lambda}_{1}{c}_{1}{\overrightarrow{v}}_{1}+{\lambda}_{2}{c}_{2}{\overrightarrow{v}}_{2}\phantom{\rule{0ex}{0ex}}\Rightarrow A\overrightarrow{u}=3{c}_{1}{\overrightarrow{v}}_{1}+2{c}_{2}{\overrightarrow{v}}_{2}\phantom{\rule{0ex}{0ex}}$

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

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

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 $\overrightarrow{u}.A\overrightarrow{u}$ is as represented below:

$-2\le \overrightarrow{u}.A\overrightarrow{u}\le 3$

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.

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