FCDiffEqChangeVariables[mat, x, y, rule, yOfX] applies a variable transformation from x to ydescribed by rule, where yOfX denotes y(x). Here mat is a matrix in the context of differential equations, i.e. it can be either the matrix \mathcal{A} or \mathcal{B} from the pre-canonical F' = \mathcal{A} F or canonical G' = \varepsilon \mathcal{B} G form, or the transformation matrix \mathcal{T} with F = \mathcal{T} G .
By default, the transformation also includes the prefactor 1/f'(y). This is correct for \mathcal{A} or \mathcal{B} but not for \mathcal{T} matrices. The inclusion of the prefactor can be disabled by setting the option Prefactor to False.
mat = {{(-2*(-1 + eps))/x, 0, 0, 0}, {0, (1 - eps)/x, 0, 0}, {0, (-2*(-1 + eps))/(x*(-1 + 4*x)),
(-2*(-1 + 2*eps))/(-1 + 4*x), 0}, {(-2*(-1 + eps))/(x*(-1 + 4*x)), 0, 0,
(-1 + eps + 6*x - 8*eps*x)/(x*(-1 + 4*x))}}\left( \begin{array}{cccc} -\frac{2 (\text{eps}-1)}{x} & 0 & 0 & 0 \\ 0 & \frac{1-\text{eps}}{x} & 0 & 0 \\ 0 & -\frac{2 (\text{eps}-1)}{x (4 x-1)} & -\frac{2 (2 \;\text{eps}-1)}{4 x-1} & 0 \\ -\frac{2 (\text{eps}-1)}{x (4 x-1)} & 0 & 0 & \frac{-8 \;\text{eps} x+\text{eps}+6 x-1}{x (4 x-1)} \\ \end{array} \right)
matNew = FCDiffEqChangeVariables[mat, x, y, x -> (1 - y^2)/4, Sqrt[1 - 4*x], Assumptions -> {y > 0}]\left( \begin{array}{cccc} -\frac{4 (\text{eps}-1) y}{y^2-1} & 0 & 0 & 0 \\ 0 & -\frac{2 (\text{eps}-1) y}{y^2-1} & 0 & 0 \\ 0 & \frac{4-4 \;\text{eps}}{y-y^3} & \frac{1-2 \;\text{eps}}{y} & 0 \\ \frac{4-4 \;\text{eps}}{y-y^3} & 0 & 0 & \frac{4 \;\text{eps} y^2-2 \;\text{eps}-3 y^2+1}{y-y^3} \\ \end{array} \right)
Setting the option Reverse to True allows to undo the transformation.
matCheck = FCDiffEqChangeVariables[matNew, x, y, x -> (1 - y^2)/4, Sqrt[1 - 4*x], Reverse -> True]\left( \begin{array}{cccc} \frac{2-2 \;\text{eps}}{x} & 0 & 0 & 0 \\ 0 & \frac{1-\text{eps}}{x} & 0 & 0 \\ 0 & -\frac{2-2 \;\text{eps}}{x-4 x^2} & \frac{2-4 \;\text{eps}}{4 x-1} & 0 \\ -\frac{2-2 \;\text{eps}}{x-4 x^2} & 0 & 0 & \frac{8 \;\text{eps} x-\text{eps}-6 x+1}{x-4 x^2} \\ \end{array} \right)
Simplify[matCheck - mat] // Flatten // Union\{0\}
FCDiffEqChangeVariables[mat, x, y, x -> (1 - y^2)/4, Sqrt[1 - 4*x], Assumptions -> {y > 0},
Prefactor -> False]\left( \begin{array}{cccc} \frac{8 (\text{eps}-1)}{y^2-1} & 0 & 0 & 0 \\ 0 & \frac{4 (\text{eps}-1)}{y^2-1} & 0 & 0 \\ 0 & -\frac{8 (\text{eps}-1)}{y^2 \left(y^2-1\right)} & \frac{4 \;\text{eps}-2}{y^2} & 0 \\ -\frac{8 (\text{eps}-1)}{y^2 \left(y^2-1\right)} & 0 & 0 & \frac{\text{eps} \left(8 y^2-4\right)-6 y^2+2}{y^2 \left(y^2-1\right)} \\ \end{array} \right)