What are Similar matrices? I know the definition of these matrices. What I want to ask is how they relate to the linear map that they represent. Before asking, I read on another post that similar matrices show up when wants to consider how two matrices with respect different bases of the same linear map are connected. Could someone give an example of how one would use this to come to the definition of a similar matrix.
 A: After playing around with matrices of linear maps, I remembered the change of basis formula. Let $M(T, (u_1,. . .,u_n), (v_1,. . .,v_n))$ denote the matrix of the linear map $T$ with respect to the basis $(u_1,. . .,u_n)$ and $(v_1,. . .,v_n)$. Let $M(T, (v_1,. . .,v_n)) =M(T, (v_1,. . .,v_n), (v_1,. . .,v_n)).$
Let $I$ denote identity map. Then the change of basis formula tells us the following.
$M(T, (u_1,. . .,u_n))$ = $A^{-1}M(T, (v_1,. . .,v_n))A$.
Where $A=M(I, (u_1,. . .,u_n), (v_1,. . .,v_n))$
I am assuming this is the reason they are called similar matrices. As in, we would expect the matrices of the same linear map to be connected in some way. The change of basis formula gives that connection. So they are similar in the sense that they represent the same linear map.
As an example: Define a linear map $T : R^2 \to R^2$ as follows. $T(x, y) = (x + 3y, 2y + 7x)$
Then,
$M(I, ((1, 3), (2, 4)), ((7, 2), (8, 3))) = \begin{bmatrix}-4.2 & -5.2\\3.8 & 4.8\end{bmatrix} = P$
$M(T, ((1, 3), (2, 4))) = \begin{bmatrix}-7 & -6\\8.5 & 10\end{bmatrix} = B$
$M(T, ((7, 2), (8, 3))) = \begin{bmatrix}-77 & -89\\69 & 80\end{bmatrix} = A$
Then we see that $B=P^{-1}AP$. Also, $M(I, ((1, 3), (2, 4)), ((7, 2), (8, 3)))$ has an inverse which is the matrix of the identity map with the bases changed. $M(I, ((1, 3), (2, 4)), ((7, 2), (8, 3)))^{-1}=M(I, ((7, 2), (8, 3)), ((1, 3), (2, 4)))$
