What is the purpose of conjugation of a rotation by an orthogonal matrix $A$

Question

In notes on Lie Theory I'm studying there is a conjugation by (and the remark) "an orthogonal matrix $A$ corresponding to an orthonormal basis compatible with the rotation." Then there is the expression:

$$A\pmatrix{\cos t&-\sin t&0\\\sin t&\cos t&0\\0&0&1}A^{-1}$$

where I can see the matrix in the brackets being conjugated is a rotation.

Is it correct to say the $A^{-1}$ takes a vector that would be rotated to the basis of the rotation matrix. It is then rotated, and the $A$ takes the resultant rotated vector back to its original basis?

Assuming this description is correct, how is $A$ determined?

Thanks

Answer 1

Your description is one correct way of describing it.

In any $3D$ rotation, there are two orthogonal directions that play the role of the $x,y$ axes and a remaining direction which plays the role of the axis of the $z$-axis, i.e. the axis of rotation (note that these three directions are mutually orthogonal). Let's call these axes the "$x$-like axis, the $y$-like axis, and the $z$-like axis" respectively.

$A$ is the transformation that maps the $x$-axis to the $x$-like axis, the $y$-axis to the $y$-like axis, and the $z$-axis to the $z$-like axis.

Answer 2

This is a way of expressing a rotation about z-axis (let name this coordinate frame with this z-axis as $\{ 1 \}$ ) in a different coordinate frame $\{ 0 \}$.

For vectors described in these two frames we have equation $^{0}v=A^{1}v$, we can calculate $A$ taking orthogonal unit vectors $i,j,k$ and expressing them with the use of their representations in frames as $\{ 0 \}$ and $\{ 1 \}$ as $[ ^0i \ ^0j \ ^0k] =A[ ^1i \ ^1j \ ^1k]$.

Hence $A=[ ^0i \ ^0j \ ^0k]{[ ^1i \ ^1j \ ^1k]}^{-1}$.