Ordering eigenvectors for basis transformation

Question

Let's say I have a matrix $H$ represented in some basis, $a$, and I'd like to transform this to be represented in a different basis, $b$. The only difference between the bases is that $b$ is a basis of eigenvectors of a reflection operator, $\hat{\sigma}$. So in theory, what I want to do is find the eigenvectors of $\hat{\sigma}$, but them into a matrix that I'll call $U$, and then compute $H_b = U^\dagger H U$. My problem is that the basis $a$ is ordered according to an order I want to preserve, but Mathematica by default sorts the eigenvectors of $\hat{\sigma}$ by the eigenvalues (which are all either +1 or -1 since this is a reflection operator). How can I keep the ordering of the eigenvectors locked to the original basis so I can indeed use this as a basis transformation without mixing up the subblocks of my existing matrix?

Here's a code example of what I want:

Ha = {{298.1, -0.7, 0., 0., 0., 0., 0., 0.}, {-0.7, 296.8, 0., 0., 0., 0., 0., 0.}, {0., 0., 298.1, -0.7, 0., 0., 0., 0.}, {0., 0., -0.7, 296.8, 0., 0., 0., 0.}, {0., 0., 0., 0., 298.1, -0.7, 0., 0.}, {0., 0., 0., 0., -0.7, 296.8, 0., 0.}, {0., 0., 0., 0., 0., 0., 298.1, -0.7}, {0., 0., 0., 0., 0., 0., -0.7, 296.8}};

Note that Ha is blocked in 4 2x2 blocks. When I change bases, I expect this to become blocked in 2 4x4 blocks.

Now, my operator $\hat{\sigma}$ can be represented as:

sigma={{0, 0, 0, 1., 0, 0, 0, 0}, {0, 0, 1., 0, 0, 0, 0, 0}, {0, 1., 0, 0, 0, 0, 0, 0}, {1., 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1.}, {0, 0, 0, 0, 0, 0, 1., 0}, {0, 0, 0, 0, 0, 1., 0, 0}, {0, 0, 0, 0, 1., 0, 0, 0}};

But then when I call Transpose[Eigenvectors[sigma]].Ha.Eigenvectors[sigma], I get a matrix that is horribly off-diagonal, which is just showing that the eigenvectors of sigma are not sorted properly. So, how do I keep everything sorted correctly?

EDITED TO ADD: The example above is just a simple example with numerics for you to play with. In my real case, the matrix dimensions and values with vary with different simulation cases. I would like a programatic way to do this sorting.

Answer 1

By inspection of the eigenvectors you get the following ordering:

u = ev[[{1, 3, 5, 6, 2, 4, 7, 8}]];
Transpose[u].Ha.u // Chop // MatrixForm