Find $T$ such that $X=TYT^{-1}$ Two matrices $X$ and $Y$ are similar if and only if there is an invertible matrix $T$ such that $X=TYT^{-1}$.
Given that $X$ and $Y$ are similar, how would you find a value of $T$?
 A: You obtain it from the Jordan normal form of both matrices. The $J$ matrix in the JNF is the same for both matrices (this is a consequence/definition of similarity).
Lets write the Jordan normal forms as 
$J = Q_X^{-1} X Q_X$
and 
$J = Q_Y^{-1} Y Q_Y$
Then you have
$T = Q_X Q_Y^{-1}$.
In case you have diagonalizable matrices $X$ and $Y$ the $J$ is a diagonal matrix containing the eigenvalues of both matrices. (Note that similar matrices need to have the same eigenvalues). In the general case, the $J$ matrix is block diagonal. 
But note that the Jordan normal forms need to have the same ordering of eigenvalues (but 
this can always be achieved with a proper permutation).
EDIT: vadim123 made an important point in the comments. Since the $Q$ matrix in the JNF is not unique (there is some flexibility choosing the rows corresponding to one Jordan block) also the resulting $T$ matrix is not unique.
A: What you need to do is find a pair of bases e=(e_1 \dots e_n) and f=(f_1 \dots f_n) such that the matrix of X with respect to e is identical to the matrix of Y with respect to f. The first thing you can do is look at the eigenvalues of X and Y: they have the same eigenvalues, so a correspondence between their eigenvectors is the first place you can look to build your bases. If the matrices are diagonalizable, then in fact this is all you have to do, as their eigenvectors will form a basis for each, and obviously you are finished.
A: By the definition of similarity, both $X$ and $Y$ have an identical Jordan Forms $J$. Jordan forms are similar to the original matrix, so $$J=Q^{-1}_XXQ_X \textrm{ and }J=Q^{-1}_YYQ_Y$$
so
$$Q^{-1}_XXQ_X=Q^{-1}_YYQ_Y$$
$$X=Q_XQ^{-1}_YYQ_YQ^{-1}_X$$
$$X=Q_XQ^{-1}_YY(Q_XQ^{-1}_Y)^{-1}$$
$$T=Q_XQ^{-1}_Y$$
The columns of $Q_X$ are made of of the eigenvectors for each eigenvalue of $X$. The same holds for $Q_Y$. (Generalized eigenvectors if algebraic multiplicity for an eigenvalue is greater than 1)
