# Thread: [SOLVED] Projection matrix property

1. ## [SOLVED] Projection matrix property

Can anyone explain this property to me please?

If P is an orthogonal projection matrix, then I - 2P is unitary.

Thanks.

2. Originally Posted by Bucephalus
Can anyone explain this property to me please?

If P is an orthogonal projection matrix, then I - 2P is unitary.

Thanks.

Let's see: $(I-2P)^{*}=I^{*}-2P^{*}=I-2P^{*}=I-2P$ , since a projection is orthogonal iff it is self-adjoint, and then:

$(I-2P)(I-2P)^{*}=(I-2P)(I-2P)==I-4P+4P^2=I$ (why?) $\Longrightarrow I-2P$ is unitary.

Tonio

3. that's awesome.
Thanks Tonio