Gram-Schmidtsches Orthogonalisierungsverfahren

Länge[v_] := 
      Sqrt[(∑_(k = 1)^(Length[v]))v[[k]]^2]

Clear[GramSchmidt]
GramSchmidt[A__] := 
      Module[{v, w, k, 
          m}, 
v = {A}; 
Do[
w[m] = 
            v[[m]] - (∑_(k = 1)^(m - 1))(v[[m]] . w[k])*
                  w[k]; 
w[m] = 
            w[m]/Länge[w[m]], {m, 1, 
            Length[v]}]; 
Table[
          w[k],  {k, 1, Length[v]}]
]

Symmetrische und orthogonale Matrix A

A = (1/15) {{10, 5, 10}, {5, -14, 2}, {10, 
            2, -11}}

Determinante

Det[A]

Charakteristisches Polynom und Eigenwerte

CharacteristicPolynomial[A, t]

CharacteristicPolynomial[A, t] // Factor

Eigenvalues[A]

sys = Eigensystem[A]

Table[B[k ] = sys[[2, k]], {k, 3}]

gs = GramSchmidt[B[1], B[2], B[3]]

orthS = 
      Transpose[{gs[[1]], gs[[2]], 
          sys[[2, 3]]/Norm[sys[[2, 3]]]}]

Inverse[orthS]

Transpose[orthS]

Inverse[orthS] == Transpose[orthS]

Transpose[orthS] . A . orthS

Transpose[orthS] . A . orthS // Simplify