Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 $ℝ^{2}$ . From the spectral theorem, we know that there exists an orthonormal eigen basis ${\vec{v}}_{1}, {\vec{v}}_{2}$ for T , with associated real eigenvalues $λ_{1} = 3$ and $λ_{2} = - 2$ (Arrange things so that $|λ_{1}| \geq |λ_{2}|$ ). Now consider the unit vector $\vec{u}$ is represented below:

$\vec{u} = c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} \Rightarrow A \vec{u} = λ_{1} c_{1} {\vec{v}}_{1} + λ_{2} c_{2} {\vec{v}}_{2} \Rightarrow A \vec{u} = 3 c_{1} {\vec{v}}_{1} + 2 c_{2} {\vec{v}}_{2}$

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

role="math" localid="1659612634271" $\vec{u} . A \vec{u} = (c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2}) . (3 c_{1} {\vec{v}}_{1} - 2 c_{2} {\vec{v}}_{2}) \Rightarrow \vec{u} . A \vec{u} = 3 c_{1}^{2} - 2 c_{2}^{2} (1)$

Since

$3 c_{1}^{2} - 2 c_{2}^{2} \leq (3 c_{1}^{2} + 3 c_{2}^{2} = 3) (2)$

and

$(- 2 c_{2}^{2} - 2 c_{2}^{2} = - 2) \leq 3 c_{1}^{2} - 2 c_{2}^{2} (3)$

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

$- 2 \leq \vec{u} . A \vec{u} \leq 3$

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

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

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Symmetric matrix:

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

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

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Linear Algebra With Applications

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Short Answer

If A is any symmetric 2x2 matrix with eigenvalues -2 and 3, and u→ is a unit vector ℝ2,ℝ2 , what are the possible values of the dot product u→×Au→ ? Illustrate your answer, in terms of the unit circle and its image A .

Step by Step Solution

Step 1: Symmetric matrix:

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

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

Most popular questions for Math Textbooks

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Pure Maths

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