Exercise  3 Section 6.3 in G&S
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Solve[Det[{{1-2p,2p,0},{p,1-2p,p},{0,2p,1-2p}}-lambda{{1,0,0},{0,1,0},{0,0,1}}]==0,{lambda}]

Solve[{{1-2 p,2 p,0},{p,1-2 p,p},{0,2 p,1-2 p}}.{{b11},{b12},{b13}}=={{b11},{b12},{b13}},{b11,b12,b13}]

Solve[{{1-2 p,2 p,0},{p,1-2 p,p},{0,2 p,1-2 p}}.{{b21},{b22},{b23}}==(1-2p){{b21},{b22},{b23}},{b21,b22,b23}]

Solve[{{1-2 p,2 p,0},{p,1-2 p,p},{0,2 p,1-2 p}}.{{b31},{b32},{b33}}==(1-4p){{b31},{b32},{b33}},{b31,b32,b33}]

B={{1,1,1},{1,0,-1},{1,-1,1}};

Simplify[B.{{1,0,0},{0,1-2p,0},{0,0,1-4p}}.Inverse[B]]

FullSimplify[B.{{1,0,0},{0,1-2p,0},{0,0,1-4p}}^n.Inverse[B]]