How to prove that the product of two projection matrices is zero?

Question

I now have $C_1$ and $C_2$, both are $n\times{n}$ symmetric, idempotent matrices and $C_1+C_2=I_n$. How do I prove that $C_1{C_2}=0$?

I'm supposed to use $a^T C_1 a- a^T C_1 a=0$ (for an arbitrary matrix $a$) to do the process.

I know I can have

$a^T C_1 a-a^T C_1 C_1 a=0$
$a^T C_1 a (I_n-C_1) =0 $
$a^T C_1 a C_2 =0 $
$a^T C_1 C_1 a C_2 C_2=0 $

But how can I get

$a^T (C_1 C_2)(C_2 C_1) a =0$

from above so I can have

$(C_1 C_2) (C_1 C_2)^T=0$

and thus conclude that

$C_1{C_2}=0$ ?
Thank you!

Answer 1

Since $C_1$ and $C_2$ are symmetric, $C_1C_2=C_2C_1$; and since they are idempotent,

$$I_n=(C_1+C_2)^2=C_1^2 +C_1C_2+C_2C_1+C_2^2 = (C_1+C_2) + 2C_1C_2 = I_n + 2C_1C_2.$$

Thus, $C_1C_2=0$.