What is the purpose of conjugation of a rotation by an orthogonal matrix $A$ 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
 A: 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.
A: 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 $A=[ ^0i \ ^0j \ ^0k]{[ ^1i \ ^1j \ ^1k]}^{-1}$.
