Eigenvalues and eigenspaces of a matrix with a given property
Suppose
- Show that the only possible eigenvalues of
are 0 and 2. - For each
, let denote the -eigenspace of , i.e., . Prove that . (Hint: For every vector one can write .)
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Suppose $A$ is a real $n\times n$ matrix that satisfies $A^2 {\bf v} = 2A{\bf v}$ for every ${\bf v}\in {\bf R}^n$.
\begin{enumerate}[label=\alph*)]
\item Show that the only possible eigenvalues of $A$ are 0 and 2.
\item For each $\lambda\in {\bf R}$, let $E_{\lambda}$ denote the $\lambda$-eigenspace of $A$, i.e., $E_{\lambda} = \{{\bf v}\in {\bf R}^n\mid A{\bf v}=\lambda {\bf v}\}$. Prove that ${\bf R}^n = E_0\oplus E_2$. ({\itshape Hint:} For every vector ${\bf v}$ one can write ${\bf v}=({\bf v}-\frac{1}{2}A{\bf v})+\frac{1}{2}A{\bf v}$.)
\end{enumerate}