Is this projection matrix?

Question

What we know about projection matrices:

$A^T=A $

$A=A^2 $

I want to prove that $(I-A)^T $is also projection matrix :

My solution:

$(I-A)^T =I-A^T=I-A=I-A^2$

we can see that $ I-A=I-A^2 => 0=A-A^2$, $A^2=A$ then $A-A=0$

Does it look correct?

Thanks for help!

Answer 1

Hint:

You need to prove that $$\left((I-A)^T\right)^T = (I-A)^T, \quad \left((I-A)^T\right)^2 = (I-A)^T.$$

Answer 2

I don't understand what you mean by "we can see that $ I-A=I-A^2 => 0=A-A^2$ $A^2=A$ then $A-A=0$".

However

$$\begin{aligned} \left(I - A^T\right)^2 & = (I-A)^T(I-A)^T\\ &=\left((I-A)(I-A)\right)^T\\ &= (I-2A + A^2)^T\\ &=(I-A)^T \end{aligned}$$

So $(I-A)^T$ which satisfies the two required identities to be a projection is indeed a projection.