Let $A\in M_{n \times n}(\mathbb{C})$ and $A^2 =-1$. Determine if A is diagonalizable or not?
A $ n \times n$ matrix A is called diagonalizable if it either has n distinct eigenvalues and if the eigenvalues are not distinct then the eigenspace has dimension n.
So, I tried finding the eigen values first so, using $|A-\lambda I|=0$ to find $\lambda$ but I am not able to make any progress and I am unable to understand how should I use $A^2 = -1$ here.
Can you please help?
Best Answer
I suppose you mean $A^2=-I$, where $I$ denotes the identity matrix. Then $X^2+1$ splits with simple roots and annihilates $A$, hence $A$ is diagonalisable.