# Change of coordinates of a matrix $A$ to a basis formed by its eigenvectors

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?

It's not the matrix that has a diagonal representation, but the linear map $A\vec x$. If we change the basis of the vector space, the coordinates of any point will change. Hence, the matrix which represents the linear map will change.
• They don't use this statement in wikipedia article. My last question was unrelated to a different article. See the 'Base of eigenvectors' part, first paragraph. What is a diagonal representation of matrix $A$ there? – user216094 Mar 20 '15 at 18:55
• The statement is not well written. Start start with a linear transformation which is represented by a matrix $A$ using the standard basis. Now, find the eigenvalues and eigenvectors of the linear map. These are the same as the eigenvalues and eigenvectors of the matrix. Now expressing the same linear map using the basis of eigenvectors will represent the linear map as a diagonal matrix. – Tim Raczkowski Mar 20 '15 at 21:36