Kunze Linear Algebra · Problem 7.2.16

Rational and Jordan Forms

Mathematics — Methods · matrix-equation

Problem

Let

A

be an

n \times n

matrix with real entries such that

A^2 + I = 0

. Prove that

n

is even, and if

n = 2k

, then

A

is similar over the field of real numbers to a matrix of the block form

\begin{bmatrix} 0 & -I \\ I & 0 \end{bmatrix}

where

I

is the

k \times k

identity matrix.