The IsSimilar(A, B) command returns false if A and B are not similar, and otherwise returns the expression sequence true, P, where P is the similarity transformation matrix, that is, $A=P^{\left(\mathrm{-1}\right)}·B·P$.

If either of the Matrices has a floating-point data type, then each must evaluate to a purely floating-point Matrix. In this case, the result of IsSimilar is achieved by making a numeric comparison of the eigenvalues of each Matrix.

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

$\mathrm{IsSimilar}\left(⟨⟨1\,2⟩|⟨3\,4⟩⟩\,⟨⟨9\,17⟩|⟨-2\,-4⟩⟩\right)$

$\mathrm{true},\left[\begin{array}{cc}1& 0\\ 4& -\frac{3}{2}\end{array}\right]$

$A≔⟨⟨3\,1⟩|⟨2\,2⟩⟩$

$A≔\left[\begin{array}{cc}3& 2\\ 1& 2\end{array}\right]$

$B≔\mathrm{DiagonalMatrix}\left(\left[1\,4\right]\right)$

$B≔\left[\begin{array}{cc}1& 0\\ 0& 4\end{array}\right]$

$S≔\mathrm{IsSimilar}\left(A\,B\right)$

$S≔\mathrm{true},\left[\begin{array}{cc}-\frac{1}{3}& \frac{2}{3}\\ \frac{2}{3}& \frac{2}{3}\end{array}\right]$

${S\left[2\right]}^{-1}·B·S\left[2\right]$

$\left[\begin{array}{cc}3& 2\\ 1& 2\end{array}\right]$