Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

If A is any symmetric 3x3 matrix with eigenvalues -2,3 , and 4 , and $\vec{u}$ is a unit vector in $ℝ^{3}$ , what are the possible values of the dot product $\vec{u} . A \vec{u}$ ?

The possible values of the dot product $\vec{u} . A \vec{u}$ are $- 2 \leq \vec{u} . A \vec{u} \leq 4$ , where is a unit vector

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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 3x3 matrix with eigenvalues -2,3 and 4 , and $\vec{u}$ is a unit vector in $ℝ^{3}$ . From the spectral theorem, we know that there exists an orthonormal eigenbasis ${\vec{v}}_{3} f o r T$ , with associated real eigenvalues $λ_{1} = 4, λ_{2} = 3$ and $λ_{3} = - 2$ (Arrange things so that $|λ_{1}| \geq |λ_{2}| \geq |λ_{3}|$ ). Now consider unit vector as represented below:

$\vec{u} = c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + c_{3} {\vec{v}}_{3} \Rightarrow A \vec{u} = λ_{1} c_{1} {\vec{v}}_{1} + λ_{2} c_{2} {\vec{v}}_{2} + λ_{3} c_{3} {\vec{v}}_{3} \Rightarrow A \vec{u} = 4 c_{1} {\vec{v}}_{1} + 3 c_{2} {\vec{v}}_{2} + 2 c_{3} {\vec{v}}_{3}$

Now evaluate as follows:

$\vec{u} . A \vec{u} = (c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + c_{3} {\vec{v}}_{3}) . (4 c_{1} {\vec{v}}_{1} + 3 c_{2} {\vec{v}}_{2} + 2 c_{3} {\vec{v}}_{3}) \Rightarrow \vec{u} . A \vec{u} = 4 c_{1}^{2} + 3 c_{2}^{2} - 2 c_{3}^{2} (1)$

Since

$4 c_{1}^{2} + 3 c_{2}^{2} - 2 c_{3}^{2} \leq (4 c_{1}^{2} + 4 c_{2}^{2} - 4 c_{3}^{2} = 4) (2)$

and

$(- 2 c_{1}^{2} - 2 c_{2}^{2} - 2 c_{3}^{2} = - 2) \leq 4 c_{1}^{2} + 3 c_{2}^{2} - 2 c_{3}^{2} (3)$

From (1),(2) and (3) we can imply that the possible values of the dot product areas represented below:

$- 2 \leq \vec{u} . A \vec{u} \leq 4$

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Linear Algebra With Applications

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Short Answer

If A is any symmetric 3x3 matrix with eigenvalues -2,3 , and 4 , and $\vec{u}$ is a unit vector in $ℝ^{3}$ , what are the possible values of the dot product $\vec{u} . A \vec{u}$ ?

Step by Step Solution

Step 1: Symmetric matrix:

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

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Linear Algebra With Applications

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Short Answer

If A is any symmetric 3x3 matrix with eigenvalues -2,3 , and 4 , and u→ is a unit vector in ℝ3 , what are the possible values of the dot product u→.Au→ ?

Step by Step Solution

Step 1: Symmetric matrix:

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

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If A is any symmetric 3x3 matrix with eigenvalues -2,3 , and 4 , and $\vec{u}$ is a unit vector in $ℝ^{3}$ , what are the possible values of the dot product $\vec{u} . A \vec{u}$ ?