# Eigenvectors of Hermitian matrices [duplicate]

I asked a similar question in the physics stack exchange, but realized my question is probably more suited here.

For any Hermitian matrix $$H = H^{\dagger}$$ we can write $$H = P DP^{\dagger}$$ where $$P$$ is the matrix containing the eigenvectors and $$D$$ is the matrix containing the eigenvalues on the diagonal. In this case $$P^{-1} =P^{\dagger}$$ and $$P^{\dagger}$$ means the Hermitian conjugate.

So I wanted to do this for a specific matrix of the form

h = {{x, -d}, {- d, -x}}


where $$x$$ and $$d$$ are both real.

I wrote

Eigensystem[h]


and obtained

{-(x+ Sqrt[d^2 + x^2])/d,1} and {(-x+ Sqrt[d^2 + x^2])/d,1}


as eigenvectors and

Sqrt[d^2 + x^2] and -Sqrt[d^2 + x^2]


as the corresponding eigenvalues.

However, if we now use the eigenvectors to form the matrix $$P$$

p = {
{-(x+ Sqrt[d^2 + x^2])/d,(-x+ Sqrt[d^2 + x^2])/d },
{1,1}
}


we see that $$P$$ does not satisfy $$P^{-1} \neq P^{\dagger}$$.

In physics I think this is a big deal, because it means that your eigenstates does not satisfy the necessary (anti)commutation relations.

My question is: How can I get mathematica to give me the eigenvectors that makes sure $$P^{-1} = P^{\dagger}$$?

I'm asking this question because I eventually want to generalize this procedure to a hermitian block diagonal $$8 \times 8$$ matrix.

In my case I think my normalized eigenvectors should be $$(u,v)$$ and $$(v,u)$$ where

$$u^2 = \frac{1}{2}\left(1 + \frac{\xi}{\sqrt{\xi^2 + \Delta^2}} \right)$$, $$v^2 = \frac{1}{2}\left(1 - \frac{\xi}{\sqrt{\xi^2 + \Delta^2}} \right)$$

However, I can't obtain this by just writing

FullSimplify[Normalize[{-(x+ Sqrt[d^2 + x^2])/d,1}]]


Is there an easy way to simplify this using Mathematica?

• Normalize the eigenvectors. (This is done on approximate numeric matrices by Eigensystem[] but not on symbolic or exact matrices. See "Details and Options" of the documentation for Eigensystem, 2nd & 3rd points.) Commented Jun 25, 2019 at 17:51
• Related: (72941), (79862), (139216), (151547) Commented Jun 25, 2019 at 18:01