What does it mean to change coordinates of a matrix? A matrix is just a bunch of numbers. All we can do is try to experss it as a linear combination of some other matrices of the same size.
I don't quite understand this article. It is shown that a change of coordinates of a matrix $A$ to a basis formed by its eigenvectors results in a diagonal matrix. I know what eigenvalues and eigenvectors are, I understand matrix diagonalization, but all I can follow here is the first two equations. What does this new diagonal matrix represent with respect to the original matrix $A$? Does it have anything to do with matrix diagonalization? I guess not.
What do we mean by 'matrix $A$ has a diagonal representation'? The possiblity of diagonalization of the matrix?