Determine eigenvalues and eigenvectors of a linear map

Question

Let $\underline v$ be a fixed vector such that $|\underline v|=1$ and let $$ T:E_3\longrightarrow E_3,\qquad T(\underline x)=(\underline x\times \underline v)\times\underline v. $$ Then

1. All possible eigenvalues of $T$ are greater than $0$
2. No one of other answers
3. All possible eigenvalues of $T$ are less or equal than $0$
4. All possible eigenvalues of $T$ are less than $0$
5. All possible eigenvalues of $T$ are greater or equal than $0$

My attempt. By definition, $\lambda$ is an eigenvalue of $T$ if there exists $\underline x\neq\underline 0$ such that $$ T(\underline x)=(\underline x\times\underline v)\times\underline v=\lambda x. $$ This implies that $\lambda\underline x\cdot(\underline x\times\underline v)=0$, which holds if and only if $\lambda=0$. Then $$ (\underline x\times\underline v)\times\underline v=\underline0 $$ if and only if $\underline x=k\underline v$, $k\in\mathbb{R}$ (which are the eigenvectors relative to $\lambda=0$). So the correct answer is 2.

Is my attempt right?

Thank You

Answer 1

The identity $\lambda {\bf x} \cdot ({\bf x} \times {\bf v}) = 0$ holds for all ${\bf x}, {\bf v} \in E_3$, because ${\bf x} \cdot ({\bf x} \times {\bf v}) = 0$ does---thus we cannot conclude anything about $\lambda$ from this identity.

Hint One option here is to use the dot product expression for the "triple cross product": We have $$T({\bf x}) := ({\bf x} \times {\bf v}) \times {\bf v} = -(\underbrace{{\bf v} \cdot {\bf v}}_1) {\bf x} + ({\bf v} \cdot {\bf x}) {\bf v} = - {\bf x} + ({\bf v} \cdot {\bf x}) {\bf v} .$$ So, if $\bf x$ is an eigenvector of $T$ of eigenvalue, say, $\lambda$, using the definition and rearranging gives $$({\bf v} \cdot {\bf x}) {\bf v} - (\lambda + 1) {\bf x} = 0 .$$ Can you take it from here?

Additional hint Consider separately the cases that ${\bf v}$ and ${\bf x}$ are (1) linearly dependent, and (2) linearly independent.

Alternatively, we can recognize our above formula for $T$ as the formula for the orthogonal projection of $-\bf x$ onto the plane orthogonal to $\bf v$.