The Similarity Transform: Things I hadn't noticed

As soon as I got back into graduate physics, I started noticing transforms of matrix operators that looked like this:

A is the original matrix operator A prime is the matrix operator transformed by gamma.  Gamma is any kind of vector transformation.  It might be a rotation, or a change of coordinate system, (from Cartesian to polar for example)..  Presented in this manner, the origins of the transform, A acting on gamma and the product acted upon by the inverse of gamma didn't make any sense to me.  I found an article, (I'll try to get a reference up here soon), that gave a very detailed very academic explanation, but it was still no good for me.  Recently, a professor finally went through the steps that arrive at the above.  It was short concise, and made sense!  Here they are.

Gamma is a matrix that transforms a vector into another vector, say... x prime into x.  I mentioned that already.

The inverse of gamma will convert an x vector into an x prime vector.

A is defined to be a matrix that operates on an x vector and returns a y vector.

Suppose we want to transform A so that it can operate on x prime vectors and return y prime vectors like this.

We can rewrite the x vector as an x prime vector like this:

.

The next step is to change the y vector into a y prime vector.

And we're done.  From here, you can see that since we wanted to get

then

Like I said, there's a deeper meaning to all of this, but now I know the simple steps that lead to the end result.