(* Gamma 0 and Gamma 1 *)
G0 = {{2(c^2)+(su^2),0},{0,(su^2)}};
G1 ={{-(c^2),0},{(su^2),0}};
Omg={{o11,o12},{o21,o22}}; (* the matrix Omega *)

(* this is the equation we need to solve for Omega *)
OmegaVals=Solve[Flatten[Omg+G1.Transpose[Inverse[Omg]].Transpose[G1]-G0,1]=={0,0,0,0},{o11,o12,o21,o22}]

OmegaSol={{o11,o12},{o21,o22}}/.OmegaVals[[1]];
MatrixForm[((c^2)+(su^2))OmegaSol]

(* solve for theta_1 *)
T=G1.Inverse[OmegaSol]//Simplify;
MatrixForm[T]

(* check that these agree with the book *)
Together[ 2(c^2)+(su^2)-((c^4)/((c^2)+(su^2)))]

(* compute powers of the matrix theta_1 *)
T
T.T//Simplify
T.T.T//Simplify
T.T.T.T//Simplify