Suppose that $V$ is a finite dimensional vector space and $T \in L(V)$. Let $B$ be a basis for $V$ such that $[T]_B = A$. Let $A$ be similar to a matrix $D$. Then does there exist any basis $B'$ for $V$ such that $[T]_{B'} = D$?
If the answer is 'yes' how can the basis $B'$ for $V$ be determined? Please help me.
Thank you in advance.
Yes.
If $A=XDX^{-1}$, then $X$ converts between coordinates with respect to $B$ and $B'$.