You should use the columns of the matrix for the arrows, not the rows:

A = {{1, 1, 1}, {1, 6, 8}, {1, 8, 9}};
c = Orthogonalize[Eigenvectors[A]] // N;
c // MatrixForm
Graphics3D[{GeometricTransformation[Cuboid[], c], 
  Map[Arrow[{{0, 0, 0}, #}] &, Transpose[1.5 c]]}, Axes -> True]

or else

Graphics3D[{GeometricTransformation[{Cuboid[], 
    Map[Arrow[{{0, 0, 0}, #}] &, 1.5 IdentityMatrix[3]]}, c]}, 
 Axes -> True]

Thank you for your response, Gianluca.

I tried to do what you show me, but I couldn't get the result I am looking for. I show the code and a picture with my code and the result I obtained...maybe someone find the mistake.

A = {{1, 1, 1}, {1, 6, 8}, {1, 8, 9}};
c = Orthogonalize[Eigenvectors[A]] // N;
c // MatrixForm
Graphics3D[{GeometricTransformation[Cuboid[], c], 
  Arrow[{{0, 0, 0}, {0.09489129421592486, 0.636336138762876, 
     0.7655533689994319}}], 
  Arrow[{{0, 0, 0}, {-0.9918275784421267, 0.005450756929915728, 
     0.12746899187848795}}], 
  Arrow[{{0, 0, 0}, {0.08528597143524745, -0.7713926417546674, 
     0.6306185022048604}}]}, Axes -> True]

I'd wish the cuboid coincide with the arrows.

Thank you.

I'd rather not make assumptions about what you mean by "change the reference system". Maybe you can provide a before and after comparison so I know what you're trying to change.