Calculating an exact orthogonal modal matrix in Mathematica

Question

I can calculate the modal matrix of a matrix A using the command JordanDecomposition[A][[1]], and a decimal approximation of the orthogonal (or normalised) modal matrix using JordanDecomposition[N[A]][[1]] but I would like to know how to calculate the answer in an exact form.

The example I'm using is A = {{1, 1/2, -(1/2)}, {1/2, 1, 1/2}, {-(1/2), 1/2, 1}} where using the method above I get a orthogonal modal matrix of

{{0.57735, -0.57735, 0.57735}, {-0.57735, 0.211325, 0.788675}, {0.57735, 0.788675, 0.211325}}

which results in, for $ y = x^T . A . x $

{{y1}, {y2}, {y3}} == {{0.57735 x1 - 0.57735 x2 + 
0.57735 x3}, {-0.57735 x1 + 0.211325 x2 + 
0.788675 x3}, {0.57735 x1 + 0.788675 x2 + 0.211325 x3}}

rather than the exact form: $$ y_1 = \frac{1}{\sqrt{3}} x_1 - \frac{1}{\sqrt{3}} x_2 + \frac{1}{\sqrt{3}} x_3 $$ $$ y_2 = \frac{1}{\sqrt{2}} x_2+\frac{1}{\sqrt{2}} x_3 $$ $$ y_3 = \frac{2}{\sqrt{6}} x_1 + \frac{1}{\sqrt{6}} x_2 - \frac{1}{\sqrt{6}} x_3 $$

How can I get this exact form using Mathematica commands?

Answer 1

Works nicely:

Orthogonalize[First[JordanDecomposition[{{1, 1/2, -1/2}, {1/2, 1, 1/2}, {-1/2, 1/2, 1}}]]]

   {{1/Sqrt[3], -1/Sqrt[3], 1/Sqrt[3]},
    {-1/Sqrt[2], 0, 1/Sqrt[2]},
    {1/Sqrt[6], Sqrt[2/3], 1/Sqrt[6]}}