Different representation matrices for the same linear transformation?

Question

My question is about linear transformations and their representation matrices.

Given the matrices $A, B,$ Such that:

$$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}, B= \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$

Is there a linear transformation $T:R^3 \to R^3$ such that both $A$ and $B$ are the representation matrix of $T?$

I heva no idea how to slve that problem, maybe the answer is yes because the both matrices have the same eigen values?

Is there any other way to get the solution?

Thanks for help!!

Answer 1

Let $B_1:=\{b_1,b_2,b_3\}$ be a basis of $V:=\mathbb R^3$ and define the linear mapping $T:V \to V$ by

$T(b_1)=b_1, T(b_2)=2b_2$ and $T(b_3)=3 b_3$.

Then $T$ has the representation matrix $A$.

If $B_2:=\{b_2,b_1,b_3\}$, then the representation matrix of $T$ is $B$

Answer 2

Yes the two matrices are similar and they represent the same transformation indeed

$$B=P^{-1}AP$$

$$ \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix}= \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$