How do active and passive transformations affect how a matrix is represented?

Question

Following this question: Active and passive transformations in Linear Algebra
I wanted to ask how do matrices change with each transformation and why is it so(proof or/and intuition).
Thanks!

Answer 1

Say you have an orthogonal transformation $T$ which brings a change in the coordinate system $S_0$ to a system $S_1$ And suppose a matrix $A$ transforms any vector $\mathbf r_0$ to $\mathbf r_0'$ in $S_0$ system

And I assume you are asking what would be the new matrix $A'$ in the system $S_1$

we can write $$\mathbf A \mathbf r_0=\mathbf r_0'$$ $$\mathbf Tr_0=\mathbf r_1$$ $$\mathbf T \mathbf r_0'=\mathbf r_1'$$ $$\mathbf A \mathbf T^{-1}\mathbf T \mathbf r_0=\mathbf r_0'$$ $$\mathbf T \mathbf A \mathbf T^{-1} \mathbf r_1=\mathbf r_1'$$

So the matrix $\mathbf A'=\mathbf T \mathbf A \mathbf T^{-1}$ in system $S_1$ is equivalent to the matrix $\mathbf A$ in $S_0$

The above is true for both active (axes fixed, vectors rotate) and passive (axes rotate, vectors fixr) transformations, the only change is that $\mathbf T$ is different in active and passive transformations

for example if the $\mathbf T$ represents active rotation in the anti-clockwise direction by an angle $\theta$ about the $z$ axis then $$ T=\begin{bmatrix} \cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\\ \end{bmatrix}$$

Fora passive transformation replace $\theta$ by$-\theta$