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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 vec...