What do you want exactly? All matrices equivalent to ?
Well, the easiest and dumbest answer is to let . One way to find a similar matrix is by making the matrix B be a matrix with the eigenvalues of as the diagonal of . Then the matrix would have the corresponding eigenvectors as columns and . That is, if is an eigenpair then
and
.
Somehow I doubt this is what you are looking for.
You said before that you wanted "any" matrix equivalent to A. As Ivleph said, take any (invertible) 3 by 3 matrix, P, find [tex]P^{-1}[/math ]and calculate .
If you specifically want a diagonal matrix equivalent to A, as you seem to be saying now, you must first prove that such at thing exists- that A is "diagonalizable" which is true if and only if A has 3 independent eigenvectors. Once you have found those eigenvectors, form the matrix P having the eigenvectors as columns. The is a diagonal matrix equivalent to A.