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 where I is k*k identity matrix.
Hint: If n=2k then there is a natural ring morphism of . Now, the image of your matrix is a complex matrix with . This implies that the minimal polynomial of divides , and thus is separable. Thus, is diagonalizable, and it has eigenvalues . Now, what?
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 A2+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!)
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