Decompose Involutory Matrix into Difference of Two Idempotents

[christang_1023]

Homework Statement

A square matrix $A$ is said to be
• an involutory matrix if $A^2 = I$,
• an idempotent if $A^2 = A$.
Show that every involutory matrix can be expressed as a difference of two idempotents.

Relevant Equations

The involutory matrix can be decomposed to $A=\frac{1}{2}(I+A)-\frac{1}{2}(I-A)$

I feel confused about proving the two terms are idempotents.

 

[Mark44]

Homework Statement: A square matrix $A$ is said to be
• an involutory matrix if $A^2 = I$,
• an idempotent if $A^2 = A$.
Show that every involutory matrix can be expressed as a difference of two idempotents.
Homework Equations: The involutory matrix can be decomposed to $A=\frac{1}{2}(I+A)-\frac{1}{2}(I-A)$

I feel confused about proving the two terms are idempotents.

You're given an involutory matrix A, which means that $A^2 = I$. Clearly A can be decomposed into $\frac{1}{2}(I+A)-\frac{1}{2}(I-A)$.

Look at each of the matrices $\frac{1}{2}(I+A)$ and $\frac{1}{2}(I-A)$ and show that each of them is an idempotent matrix.