TITLE: Let T be a linear transformation on the real vector space $\mathbb{R^n}$ over $\mathbb R$ such that $T ^2 =μT$ for same $μ∈\mathbb R$
QUESTION [5 upvotes]: Let $T$ be a linear transformation on the real vector space $\mathbb R^n$ over $\mathbb R$ such that $T^2 =\mu T$ for some $\mu\in\mathbb R$ . Then which of the following is/are true?   


$\|Tx\| = |\mu| \|x\|$ for all $x \in\mathbb {R^n}$  
If $\|Tx\| =  \| x\| $for some non zero vector $x \in\mathbb R^n$, then $\mu=\pm1$  
$T= \mu I$ where $I$ is the identity transformation on $\mathbb R^n$    
If $\|Tx \|>\|x\|$ for a non zero vector $x \in \mathbb R^n$, then $T$ is necessarily singular.   


I am completely stuck on it. Can anybody help me please?

REPLY [6 votes]: Hint: Consider $T=\begin{pmatrix}\sqrt{2}&0\\0&0\end{pmatrix}$ for items 1-3. In each of item 1 and item 2, find a suitable $x$ to refute the statement. For item 4, consider $T=2I$.