ABSTRACT
Consider how an n×n matrix A transforms each of the vectors bj. Suppose that there are vectors q and b such that
If bj=Uej, then q j=AUej. Define the vector rj by qj=Urj, or rj=U tqj. Then
Multiplying from the left by Ut,
Now define the matrix M=UtAU. Then r=Me and q=Ab represent the same linear transformation, except that the basis vectors have been changed. Therefore the transformation UtAU produces a new matrix M that represents the same linear transformation as A did, but with different basis vectors.
