0
$\begingroup$

If $2$ is the only eigen value of $A\in Mat_{n\times n}(\mathbb C)$ then what can I say about the diagonalizabilty of $A$? I tried to check the equality of algebraic & geometric multiplicity of $2$.

2 Answers 2

0

You can easily conclude that either $A=2I_n$ or $A$ is not diagonalizable.

Try to prove by yourself that if $A$ is diagonalizable then $A=2I_n$, which yields the above result. If you have troubles I can give some extra hints.

  • 0
    $A$ is diagonalizable $\Rightarrow$ $\exists$ non-singular $P$ such that $P^{-1}AP=2I$ $\Rightarrow$ $A=2I$.2012-12-09
  • 0
    @SugataAdhya Yup. So there is only one such diagonalizable matrix.2012-12-09
  • 0
    Got it. Thanks. So the result is actually true for any natural number.2012-12-09
  • 0
    @SugataAdhya Yes.2012-12-09
3

Nothing. Even with $n=2$ you may have $$A=\left(\begin{matrix}2&0\\0&2\end{matrix}\right)$$ or $$A=\left(\begin{matrix}2&1\\0&2\end{matrix}\right)$$