Manipulate[ Plot[ Sin[2 Pi b ( n + 1/2)]/Sin[ Pi b], {b, 0, 1-0.1}, PlotRange -> Full], {{n, 10}, 10, 1000, 10}]

Clear[a, l, e, d, f, h] ;

a = {{-2, 1, 1}, {1, -2, 1}, {1, 1, -2}} ;
"a"
a // MatrixForm

{l, e} = Eigensystem[ a ]  ;
d = DiagonalMatrix[l] ;
f = Transpose[Orthogonalize[ e ]] ;

"f"
f // MatrixForm

"f^-1"
h = Transpose[f]

"f . f^-1"
f . h // MatrixForm

"d = f^-1 . a . f"
h .a. f // Simplify // MatrixForm

"a = f . d . f^-1"
f . d . h // MatrixForm

"sqrt(a) = f . sqrt(d) . f^-1 * sqrt(3) i"
f . Sqrt[d] . h  (Sqrt[3]I)// Simplify// MatrixForm

"a^-1/2 = f . d^-1/2 . f^-1 "
f . DiagonalMatrix[ {l[[1]]^(-1/2), l[[2]]^(-1/2), l[[3]]^(-1/2)}] . h  // Simplify// MatrixForm


"a^2 /(-3)"
a . a / (-3) // MatrixForm