The function Issimilar(A, B) is a placeholder for the boolean valued function that returns true if A is similar to B and false otherwise.  It is used in conjunction with either mod or evala.

If called in the form Issimilar(A, B,'P'), then P will be assigned a transformation matrix such that $A=\mathrm{inverse}\left(P\right)BP$.

The call Issimilar(A, B) mod p determines if A is similar to B modulo p which is a prime integer.  The entries of A and B must have rational coefficients or coefficients from an algebraic extension of the integers modular p.

The call evala(Issimilar(A, B)) determines if A is similar to B where the entries of A and B are algebraic numbers (or functions) defined by RootOfs.

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\,3\right]\,\left[4\,5\,6\right]\,\left[7\,8\,9\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

$p≔11$

$p≔11$

$\mathrm{cA}≔\mathrm{CharacteristicPolynomial}\left(A\,x\right)\mathbf{mod}p$

$\mathrm{cA}≔x^3+7x^2+4x$

$B≔\mathrm{CompanionMatrix}\left(\mathrm{cA}\,x\right)$

$B≔\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& \mathrm{-4}\\ 0& 1& \mathrm{-7}\end{array}\right]$

$\mathrm{Issimilar}\left(A\,B\,'P'\right)\mathbf{mod}p$

$\mathrm{true}$

$P$

$\left[\begin{array}{ccc}8& 6& 8\\ 8& 2& 3\\ 1& 0& 0\end{array}\right]$

$P^{-1}·B·P\mathbf{mod}11$

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$