[Q] Operator's rotation

[A23]

Let's take a 2x2 operator-matrix A acting on a vector space V, the rotation of this operator by the rotation matrix R, is given by [math]A'=R^{-1}AR[/math].

 

Now, A can be written by elements : [math]A=(a_{ij})[/math], but we could see, and write A as a 4x1 vector B, and A' represented by B' (just the elements put in column)

 

[Q] Does B and B' are linked by a 4-D rotation in this 'B-Space' ?

(Which could be written as a tensor [math]C_{ijkl}[/math] 2x2x2x2=16 components)

[sanctus]

Just what goes through my mind, but I don't really know what the meaning of B is, I mean A has the elements of the first column being with respect of a basis vector e1 and the second column with respect to besis vector e2. Then to write B is a column is kind of defining a new space with basis vector the "concatenation" of the two from the space where A lives in. Assuming B has any sense, then the rest is easy:

 

Define P to be the projection from the A-space on the B-space so you have:

[math] B=AP[/math]

So

[math] B'=A'P=R^{-1}ARP\neq R^{-1}BR[/math]

...

Might be all wrong but this is how I see it...

[A23]

Indeed