Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix.

Question

Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix.

I'm not sure how to do this, any solutions/hints are greatly appreciated.

Answer 1

Since $A$ is involutory, we have $A^2 = I$ so $A^2 - I = 0$. This means that the minimal polynomial of $A$ divides $x^2 - 1 = (x - 1)(x + 1)$. Since $x^2 - 1$ splits into linear factors, the minimal polynomial of $A$ also splits into linear factors and thus $A$ is diagonalizable.