# Why is a projection matrix of an orthogonal projection symmetric?

I am quite new to this, so I hope you forgive me if the question is naïve. (Context: I am learning econometrics from Davidson & MacKinnon’s book “Econometric Theory and Methods”, and they do not seem to explain this; I’ve also looked at Luenberger’s optimization book that deals with projections at an a bit more advanced level, but with no luck).

Suppose that I have an orthogonal projection $\mathbb P$ with is associated projection matrix $\bf P$. I am interested in projecting each vector in $\mathbb{R}^n$ into some subspace $A \subset \mathbb{R}^n$.

Question: why does it follow that $\bf{P}=P$$^T$, that is, $\bf P$ is symmetric? What textbook could I look at for this result?

This is a fundamental results from linear algebra on orthogonal projections. A relatively simple approach is as follows. If $u_1, \ldots, u_m$ are orthonormal vectors spanning an $m$-dimensional subspace $A$, and $\mathbf{U}$ is the $n \times p$ matrix with the $u_i$‘s as the columns, then

This follows directly from the fact that the orthogonal projection of $x$ onto $A$ can be computed in terms of the orthonormal basis of $A$ as

It follows directly from the formula above that $\mathbf{P}^2 = \mathbf{P}$ and that $\mathbf{P}^T = \mathbf{P}.$

It is also possible to give a different argument. If $\mathbf{P}$ is a projection matrix for an orthogonal projection, then, by definition, for all $x,y \in \mathbb{R}^n$

Consequently,

for all $x, y \in \mathbb{R}^n$. This shows that $\mathbf{P}^T = \mathbf{P}^T \mathbf{P}$, whence