Problem #1) 
This starter has 3 variables

Clear["Global`*"]

(* Put your eigenvectors here 
THESE ARE ALL FAKE NUMBERS
*)

ev1 = {1, 2, -3};
ev2 = {1, 2, -3};
ev3 = {1, 2, -3};
(* and the three eigenvalues here 
NOTE: w2 is (w^2)  in my notation *)
sol = {{w2 -> 1}, {w2 -> 2}, {w2 -> 3}}

Part B Look at motion

values={k->1,m->1};

mode1=ev1 Exp[I w t] /.{w->Sqrt[w2]} /.sol[[1]]

mode2=ev2 Exp[I w t] /.{w->Sqrt[w2]} /.sol[[2]]

mode3=ev3 Exp[I w t] /.{w->Sqrt[w2]} /.sol[[3]]

Plot[mode1 +{-2,0,2}/.values //Re,{t,0,10}]

Plot[mode2 +{-2,0,2}/.values //Re,{t,0,10}]

Plot[mode3 +{-2,0,2}/.values //Re,{t,0,10}]

Plot[(1/2)mode1 + (1/4)mode2 + (1/4)mode3 +{-2,0,2}/.values //Re,{t,0,10}]

Problem #2,3,4,5,6)
This starter has 2 variables

Clear["Global`*"]

(* Put your eigenvectors here 
THESE ARE ALL FAKE NUMBERS
*)

ev1 = {1, 2};
ev2 = {1, -2};
(* and the three eigenvalues here 
NOTE: w2 is (w^2)  in my notation *)
sol = {{w2 -> 1}, {w2 -> 2}}

Part B Look at motion

values={k->1,m->1};

mode1=ev1 Exp[I w t] /.{w->Sqrt[w2]} /.sol[[1]]

mode2=ev2 Exp[I w t] /.{w->Sqrt[w2]} /.sol[[2]]

Plot[mode1 +{-2,2}/.values //Re,{t,0,10}]

Plot[mode2 +{-2,2}/.values //Re,{t,0,10}]

Plot[(2/3)mode1 + (1/3)mode2  +{-2,2}/.values //Re,{t,0,10}]