Let A be an n*n matrix with real entries and A^2+I=0. If n=2k then show that A is similar to the block matrix of the form$\displaystyle \left(\begin{array}{cc}0&-I\\I&0\end{array}\right)$ where I is k*k identity matrix.
Hint: If n=2k then there is a natural ring morphism of $\displaystyle \text{Mat}_n(\mathbb{R})\to \text{Mat}_k(\mathbb{C})$. Now, the image of your matrix $\displaystyle A$ is a complex matrix with $\displaystyle A^2+I=0$. This implies that the minimal polynomial of $\displaystyle A$ divides $\displaystyle x^2+1$, and thus is separable. Thus, $\displaystyle A$ is diagonalizable, and it has eigenvalues $\displaystyle \pm i$. Now, what?
Hi xixi,
The fact that the matrix is over the reals and not the complex field is critical. The 2 by 2 scalar matrix A with i on the diagonal clearly satisfies A^{2}+I=0 and is certainly not similar to a matrix of the stated form. (Since A commutes with anybody, a conjugate of A is A!)
Attached is a solution. If you have questions, post your question.