Let $V$ be a vector space. For simplicity, say finite-dimensional over the reals or the complex. A linear transformation $T \in \text{End}\,(V)$ is a projection if $T^2=T$. If $V$ is in addition an inner-product space then we can talk about the operator $T^*$ adjoint to $T$, defined by
$$\forall v,w \in V, \langle Tv,w\rangle =\langle v,T^*w\rangle$$
An adjoint exists and is unique, at least for finite-dimensional spaces (compare here). $T$ is called an orthogonal projection if in addition it is self-adjoint, $T=T^*$.
Recall that a linear operator is normal if $T^*T=TT^*$. A self-adjoint operator is clearly normal.
Is every linear projection normal? How about orthogonal projections?
Best Answer
An orthogonal projection is by definition self-adjoint and hence normal.
A non-orthogonal projection need not be normal, as the following example demonstrates.
Let $P=\left(\begin{matrix}1 & 0\\1 & 0\end{matrix}\right)$ on the real plane. A direct computation shows that $P$ is a projection, $P^2=P$. The matrix of the adjoint is $\bar{P}^t=\left(\begin{matrix}1 & 1\\0 & 0\end{matrix}\right)$, which can also be verified directly to satisfy the definition of an adjoint.
However, $P$ is not normal as $PP^*=\left(\begin{matrix}1 & 0\\1 & 0\end{matrix}\right)\left(\begin{matrix}1 & 1\\0 & 0\end{matrix}\right)=\left(\begin{matrix}1 & 1\\1 & 1\end{matrix}\right)$ while on the other hand $P^*P=\left(\begin{matrix}1 & 1\\0 & 0\end{matrix}\right)\left(\begin{matrix}1 & 0\\1 & 0\end{matrix}\right)=\left(\begin{matrix}2 & 0\\0 & 0\end{matrix}\right)$.