Determining matrix of similarity

Question

Two matrices $A$ and $B$ are similar if there is an invertible matrix $P$ such that $A=PBP^{-1}$.

If the two matrices are similar, how do you find $P$?

Answer 1

$P$ is made up of the columns of eigenvectors for each eigenvalue.

See Jordan Normal Form.

You can find many examples on MSE, for example: Find a formula in terms of k for the entries of Ak, where A is the diagonalizable matrix:

Note: This answer assumes that A is diagonalizable.

Answer 2

Every matrix over $\mathbb{C}$ has a Jordan form, and there is an algortithm to construct a matrix $M$ such that $J = M^{-1} A M$. If $A$ and $B$ are similar they have the same Jordan form. Hence we got so far $$ J = M^{-1} A M \ , \qquad J = N^{-1} B N $$ where we know $M$ and $N$ (by finding a basis of generalized eigenvectors, the "Jordanizing basis". By equating these two expressions we get $$ A = M N^{-1} B N M^{-1} = (NM^{-1})^{-1} B (NM^{-1}) $$ and thus $P = NM^{-1}$.

If the problem is defined over $\mathbb{R}$ one need to show that $P$ can be chosen such that all its coefficients are real.