Projects a vector $\mathbf{y} \in \R^{m}$ onto the column space of a tall matrix $A \in \R^{m \times n}$ with full column rank through a projection matrix $P \in \R^{m \times m}$ which results in its best approximation $\mathbf{p} \in C(A)$
$$ P=A(A^TA)^{-1}A^T \hspace{15pt} P\mathbf{y} = \mathbf{p} $$
Method which obtains the best solution $\mathbf{\hat{x}} \in \R^n$ for an overdetermined system modeled by $A^{m \times n}$, $m > n$, by projecting the expected output $\mathbf{y} \in \R^m$ onto $C(A)$.
$$ \rightarrow \begin{cases} A^TA \mathbf{\hat{x}} = A^T \mathbf{y} \hspace{115pt} \text{if A is not full column rank}\\ \mathbf{\hat{x}} = (A^TA)^{-1}A^T\mathbf{y} \text{ or } A\mathbf{x} = P \mathbf{y}\hspace{43pt}\text{if A is full column rank} \\ \end{cases} $$
Generally, if $A$ is not full column rank, we might use the pseudoinverse to find the minimum norm solution. Otherwise, we will instead produce infinitely many solutions to the system.
Square matrices $Q \in \R^{n \times n}$ which have the special characteristics that its inverse is its transpose and that its associated linear transformation is isometric (preserves inner products and norms).
$$ Q^TQ=QQ^T = I \\ \langle Q \mathbf{x}, \space Q \mathbf{y} \rangle = \langle\mathbf{x}, \space \mathbf{y} \rangle \\ \langle Q \mathbf{x}, Q \mathbf{x} \rangle = \langle \mathbf{x} , \mathbf{x}\rangle $$
These matrices are constructed from mutually orthogonal columns and rows that are all unit norm. The isometry of their transformations preserve distances and angles, meaning they either only rotate or reflect vectors.
Tall matrices $Q \in \R^{m \times n}$, $m > n$, have most of the same properties as orthogonal matrices. Unlike wide matrices, it is possible to have $n$ $m$-dimensional, mutually orthogonal columns/rows, however this only allows for the existence of a left inverse.