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Max1

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- TL;DR Summary
- A Hermitian 2x2 matrix can be diagonalized by a similarity transform with a ##\mathrm{SU}(2)## matrix. This ##\mathrm{SU}(2)## can be represented by a 2x2 Wigner D-Matrix depending on three real parameters ##(\alpha,\beta,\gamma)##. Is there an explicit formula for the parameters?

**Motivation:**

Due to the spectral theorem a complex square matrix ##H\in \mathbb{C}^{n\times n}## is diagonalizable by a unitary matrix iff ##H## is normal (##H^\dagger H=HH^\dagger##). If H is Hermitian (##H^\dagger=H##) it follows that it is also normal and can hence be diagonalized by a unitary transformation. That means

$$

U^\dagger HU=D

$$

with ##D## a diagonal matrix with the eigenvalues of ##H## on its diagonal. According to [General form for 2x2 unitary matrices] a unitary matrix can also be written as

$$

U=\mathrm{e}^{i\phi/2}S

$$

with ##S \in \mathrm{SU}(2)## (the group of two dimensional unitary matrices with ##\mathrm{det}(S)=1##). The phase cancels anyway

$$

U^\dagger HU=\mathrm{e}^{-i\phi/2}S^\dagger H\mathrm{e}^{i\phi/2}S=S^\dagger HS\,.

$$

So a unimodular (##\mathrm{det}(S)=1##) unitary matrix S exists, which diagonalizes H. Following Sakurai "Modern Quantum Mechanics", any element of ##\mathrm{SU}(2)## can be represented by

$$

D(\mathbf{n},\phi)=\exp(-i\phi\,\mathbf{n}\cdot \pmb{\sigma}/2)=S

$$

with ##\mathbf{n}\in \mathbb{R}^{3\times3}##, ##\phi\in [0,4\pi)## and ##\pmb{\sigma}## denoting the vector of Pauli matrices. From a physical point of view ##D(\mathbf{n},\phi)## can be interpreted as a rotation of a two-component spinor ##\chi## around the ##\mathbf{n}##-axis by the angle ##\phi##. The direction of ##\mathbf{n}## can be described by an azimuthal angle and a polar angle. Hence, in total three real parameters are necessary to describe ##D(\mathbf{n},\phi)##. According to Sakurai page 179 the rotation in spin-space can also be represented by

$$

D(\alpha,\beta,\gamma)=

\begin{pmatrix}

\exp^{-i(\alpha+\gamma)/2}\cos(\beta/2) & -\exp^{-i(\alpha-\gamma)/2}\sin(\beta/2)\\

\exp^{i(\alpha-\gamma)/2}\sin(\beta/2) & \exp^{i(\alpha+\gamma)/2}\cos(\beta/2)

\end{pmatrix}

$$

and especially any ##S\in SU(2)## can be represented in the upper way. This representation is also known as the Wigner D-Matrix (with ##j=1/2## in this case, compare Wikipedia "Wigner D-Matrix").

**Question:**

I would like to diagonalize a given hermitian 2x2 Matrix ##H##. How do I obtain the three angles ##(\alpha,\beta,\gamma)##? I would like to have a forumla which gives the angles explicitly in terms of the matrix elements. I need this to diagonalize a Hamiltonian in spin space which does not commute with ##S_\mathrm{z}## and hence has some off-diagonal elements in ##S_\mathrm{z}## representation.

**Motivation 2:**

A real symmetric matrix (##A=A^\mathrm{T}##) can be diagonalized by an orthogonal matrix (##Q^\mathrm{T}=Q^{-1}##). Any orthgonal 2x2 matrix can be written as

$$

Q=

\begin{pmatrix}

\cos\theta & -\sin\theta\\

\sin\theta & \cos\theta

\end{pmatrix}\,.

$$

A real symmetric matrix is necessarily of the form

$$

A=

\begin{pmatrix}

a & c\\

c & a

\end{pmatrix}

$$

by calculating the eigenvectors and equating the matrix of eigenvectors with ##Q## one finds

$$

\theta=

\begin{cases}

\frac{1}{2}\arctan\left(\frac{2c}{a-b}\right) & \text{if}\, a-b>0 \land c>0 \\

\frac{1}{2}\arctan\left(\frac{2c}{a-b}\right)+\pi/2 & \text{if}\, a-b<0 \neq c \\

\frac{1}{2}\arctan\left(\frac{2c}{a-b}\right)+\pi & \text{if}\, a-b>0 \land c<0 \\

\end{cases}

$$

In the general case and for the special cases

$$

\theta=

\begin{cases}

0 &\text{if}\, c=0 \land a-b>0 \\

\pi/2 &\text{if}\, c=0 \land a-b<0 \\

\pi/4 &\text{if}\, c>0 \land a-b=0 \\

3\pi/4 &\text{if}\, c<0 \land a-b=0

\end{cases}\,.

$$

I tried obtaining the three angles in the SU(2) case in the same way, but did not succeed in finding a nice and short representation as in the case above. Can anyone give me some pointers on where to find an explicit representation?

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