My lecture notes write that two matrices are said to be similar if there exists an $n \times n$ invertible matrix $P$ such that $B = P^{-1} AP$.
But suppose I am given two matrices, $A$ and $B$, how do you 'show'/'determine' whether they are similar? Do you just have think up a matrix $P$ or is there a systematic way of finding it?
A second question I have if I have a matrix $A$ and find its eigenvectors, and place these in my similar matrix $P$, do I have to normalise my eigenvectors so that the matrix $P^{-1} AP$ equals a diagonal matrix, whose diagonals are the eigenvalues of $A$?
Thanks.
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$\begingroup$Question 1: The answers to this question answers it a lot better than I ever could.
Question 2: You do not have to normalise the eigenvectors. If you, say, keep one eigenvector long (which makes the corresponding column in $P$ have large values), then that is made up for in $P^{-1}$, so the diagonal values in the diagonal matrix are the eigenvalues of $A$ regardless.
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