I am studying Principal Component Analysis and having a bit of problem in understand the sentence, "orthogonal matrices preserve orthogonality". I have a matrix as below:
$A = \begin{bmatrix}cos\theta & -sin\theta \\ sin\theta & cos\theta\end{bmatrix}$
I need to show that the new two axes such as $z_1$ and $z_2$ are orthogonal by showing that the matrix $A$ is an orthogonal matrix. As defined, I can say if the matrix $A^T$ is $A$'s inverse matrix, then the matrix $A$ is an orthogonal matrix. However, I don't really understand what an orthogonal matrix preserves the orthogonality means.
Preserving orthogonality means if $x$ and $y$ are orthogonal, then $Ax$ and $Ay$ are orthogonal. We measure orthogonality using the dot product $x^Ty = 0$. Thus, if $x^Ty=0$, then $$(Ax)^T(Ay) = x^T(A^TA)y = x^TIy = x^Ty =0.$$
Thus $A$ preserves orthogonality.