= Orthogonal projection matrix
A <matrix> represents a <linear projection> when applying it twice has the same effect as applying it once. For an <orthogonal projection> with the standard <Hermitian inner product> on $\mathbb C^m$, the additional condition is that the <conjugate transpose> equals the original <matrix>:
$$
P^2=P=P^\dagger.
$$
Thus $P$ is both <idempotent> and a <Hermitian matrix>. Over the <real numbers> this says $P^2=P=P^T$. Its <image of a linear map> is the retained <linear subspace>, while its <kernel of a linear map> is the <orthogonal complement>: <vectors> in the image are unchanged, and <vectors> in the kernel become zero.
For a <matrix> $A$ with <full column rank>, projecting onto its <column space> gives
$$
P=A(A^\dagger A)^{-1}A^\dagger.
$$
The invertibility of $A^\dagger A$ follows from <linear independence> of the columns. If those columns are an <orthonormal set>, $A^\dagger A=I$ and $P=AA^\dagger$. These formulas and the conditions $P^2=P=P^T$ in the real case are developed in https://www.ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/00e9c8f0eafedeab21a3d079a17ed3d8_MIT18_06SCF11_Ses2.2sum.pdf[MIT 18.06SC, Projections onto Subspaces, pp. 1–3].
For the line spanned by $e=(1,2)^T$,
$$
P=\frac{ee^T}{e^Te}
=\frac15\begin{pmatrix}1&2\\2&4\end{pmatrix},
\qquad
P(3,1)^T=(1,2)^T.
$$
In particular, $Pe=e$ and $P(2,-1)^T=0$. The <eigenvalues> are $1$ on the retained direction and $0$ on the removed direction. This exposes the meaning of an <orthogonal projection> as retaining selected components, rather than stretching them.
Symmetry refers to the specified <inner product>. In the real weighted setting, the formula becomes $P_M=A(A^TMA)^{-1}A^TM$ and satisfies $P_M^2=P_M$ and $P_M^TM=MP_M$. For the preceding weighted line example,
$$
P_M=\frac18\begin{pmatrix}4&2\\8&4\end{pmatrix}.
$$
It is not symmetric under the ordinary <transpose>, but it is an <orthogonal projection> for the <weighted Euclidean inner product>. The matrix criterion $P=P^\dagger$ should therefore be used with the standard <Hermitian inner product>, not silently transferred to a different geometry.
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