Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Recall that a real square matrix A is called skew symmetric if $A^{T = - A}$ .
a. If A is skew symmetric, is $A^{2}$ skew symmetric as well? Or is $A^{2}$ symmetric?
b. If is skew symmetric, what can you say about the definiteness of $A^{2}$ ? What about the eigenvalues of $A^{2}$ ?
c. What can you say about the complex eigenvalues of a skew-symmetric matrix? Which skew-symmetric matrices are diagonalizable over $ℝ$ ?

Therefore the solution is

a. $A^{2}$ is symmetric.

b. $A^{2}$ is a negative definite.

c. The zero matrix is the only skew-symmetric matrix that is diagonalizable over $ℝ$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the transpose of a symmetric matrix is the same.

a.) A symmetric matrix is equal to its transpose

$A = A^{T}$

and a skew symmetric matrix is a matrix whose transpose is equal to its negative

$A^{T} = - A$

Now, if is a skew symmetric matrix, then as we just mentioned: $A^{T} = - A$ , now to check if $A^{2}$ would be skew symmetric as well, we have:

$(A^{2}) = {(AA)}^{T} = A^{T} A^{T} = (- A) (- A) = A^{2}$

therefore, $A^{2}$ is not skew symmetric, it is just symmetric.

Step 2: Determine the certainty of A2

b.) We have

$\begin{array}{l} q (\vec{x}) = \vec{x} A^{2} \vec{x} = {\vec{x}}^{T} A^{2} \vec{x} \\ = {\vec{x}}^{T} AA \vec{x} \end{array}$

from part (a) we mentioned A that is said to be a skew matrix if $A^{T} = - A$ , so

$\begin{array}{l} - {\vec{x}}^{T} (- A^{T}) A \vec{x} \\ = (A \vec{x}) . (A \vec{x}) \\ = {||A {\vec{x}}^{T}||}^{2} \\ \leq 0 \end{array}$

therefore, for all $x . q (\vec{x}) \leq 0$ , which implies that $A^{2}$ is a negative definite (its eigenvalues are less than or equal to 0).

Step 3: Determine the skew symmetric matric

c.) Let be an eigenvector corresponding to the eigenvalue,

$Av = λv$

multiply both sides by $v^{- T}$

$v^{- T} Av = {λv}^{- T} v$

note: the bar" -" above v is for complex conjugation.

$v^{- T} Av = λ {||v||}^{2}$

for the left-hand side suppose we have column vectors of the same size, a and b, then $a^{T} b$ is a $1 \times 1$ matrix which we can think of as a scalar. Now, taking transposes, since $a^{T} b$ is $1 \times 1$ , it is its own transpose, so

$a^{T} b = {(a^{T} b)}^{T} = b^{T} {(a^{T})}^{T} = b^{T} a$

here let $a = v$ and localid="1659623457872" $b = Av$ , then for our left-hand side

$\begin{array}{l} {(Av)}^{T} \bar{v} = λ {||v||}^{2} \\ v^{T} A^{T} \bar{v} = λ {||v||}^{2} \end{array}$

since A is skew-symmetric, we have $A^{T} = - A$ . Substituting,

$- v^{T} A \bar{v} = λ {||v||}^{2}$

taking the conjugate of $Av = λv$ yields $A \bar{v} = λ \bar{v}$ , notice we can substitute this into the left-hand side above

$- v^{T} λ \bar{v} = λ {||v||}^{2} - λ {||v||}^{2} = λ {||v||}^{2} - λ = λ$

Let $λ = a + ib$ ,

$- a + ib = a + ib$ which implies $a = 0$ and $λ = bi$ . We also conclude that the zero matrix is the only skew-symmetric matrix that is diagonalizable over $ℝ$ .

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Linear Algebra With Applications

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Step by Step Solution

Step 1: Determine the transpose of a symmetric matrix is the same.

Step 2: Determine the certainty of A2

Step 3: Determine the skew symmetric matric

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Linear Algebra With Applications

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Step by Step Solution

Step 1: Determine the transpose of a symmetric matrix is the same.

Step 2: Determine the certainty of A2

Step 3: Determine the skew symmetric matric

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