ClearAll[ g, ej, ej, a, b, acap, bcap, m, display, e]
g = 1/(1 - rho^2);
ej = Exp[I phij ];
ei = Exp[I phii ];
a = (ej -rho^2/ej) ei ;
acap = ((1/ej) - ej rho^2)/ei ;
b = -rho(ej - 1/ej)/ei ;
bcap = rho(ej - 1/ej) ei ;
m = g {{a, b}, {bcap, acap}};
display = {phii -> (Subscript[ϕ, i]),phij -> (Subscript[ϕ, j]), rho -> (Subscript[ρ, ij])} ;
(m /. display)// MatrixForm
e = Eigensystem[m];

ClearAll[ eigenvalues, eigenvectors, e1, e2 ]
eigenvalues = e[[1]]// FullSimplify;
eigenvectors = e[[2]] // FullSimplify;
e1 = eigenvectors[[1]];
e2 = eigenvectors[[2]];

eigenvalues // FullSimplify/. display // ColumnForm // TraditionalForm

e1 /. display // MatrixForm // TraditionalForm
e2 /. display // MatrixForm // TraditionalForm

ClearAll[out]
(*out = {0,psi ei};
(*m. out /. display /. psi ->ψ// Matrixform*)
(MatrixPower[m,n].out //FullSimplify) /. display /. psi-> ψ// MatrixForm // TraditionalForm*)

(*e1.e1 // FullSimplify /. display
e1.e2 // FullSimplify /. display
e2.e2 // FullSimplify /. display
*)