Suppose we are given the matrix $\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}\cos(\omega t)& -\sin(\omega t)\\\sin(\omega t)& \cos(\omega t)\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}$
In other words the new coordinate system is a rotating coordinate system.
Is there a matrix-based way of finding $\begin{pmatrix}\dot{x}\\\dot{y}\end{pmatrix}$ and $\dot{x}^2+\dot{y}^2$ in terms of the new coordinates?
I can, by inspection, see that the latter should be $(\dot{x'}-\omega y')^2+(\dot{y'}+\omega x')^2$ but I would appreciate a way using matrices. Also if such a matrix to "change coordinates" exist, I would appreciate an intuitive explanation of what it is doing.
Many thanks!