m={{0.75, -0.612372, 0.25}, {0.612372, 0.5, -0.612372}, {0.25, 0.612372, 0.75}}                                                                         

Eigensystem[m]                                                          

Out[2]= {{1., 0.5 + 0.866025 I, 0.5 - 0.866025 I}, 
 
                            -16
>    {{0.707107, -2.01185 10   , 0.707107}, 
 
                  -17                                     -17
>     {-9.71445 10    + 0.5 I, 0.707107 + 0. I, 9.71445 10    - 0.5 I}, 
 
                  -17                                     -17
>     {-9.71445 10    - 0.5 I, 0.707107 + 0. I, 9.71445 10    + 0.5 I}}}

Note: the eigenvalue=1 is as desired and then v={1/Sqrt[2],0,1/Sqrt[2]}

Tr[m]                                                                   

Out[3]= 2.

Solve[2==1+2 Cos[theta],theta]                                          

                                         -Pi
Out[6]= {{theta -> ConditionalExpression[--- + 2 Pi C[1], C[1] ∈ Integers]}, 
                                          3
 
                                     Pi
>    {theta -> ConditionalExpression[-- + 2 Pi C[1], C[1] ∈ Integers]}}
                                     3

Note: Cos[Pi/3]=1/2, so 60 deg solves

MatrixPower[m,6]                                                        

                              -6            -6
Out[8]= {{0.999998, 1.30708 10  , 1.60085 10  }, 
 
                 -6                      -6
>    {-1.30708 10  , 0.999997, 1.30708 10  }, 
 
                -6             -6
>    {1.60085 10  , -1.30708 10  , 0.999998}}

Note: this results is essential the unit matrix as six 60 deg rotations = unitity

Transpose[m].m                                                          

                              -17            -7
Out[9]= {{0.999999, 2.77556 10   , 5.33616 10  }, 
 
                -17                            -7
>    {2.77556 10   , 0.999999, 0.}, {5.33616 10  , 0., 0.999999}}

Note: rotational matrices are "orthogonal" meaning Transpose is Inverse...checked
