Diagonalization of 8x8 matrix with Euler angles

1. May 18, 2014

Trifis

I am trying to diagonalize the following matrix:
$M = \left( \begin{array}{cccc} 0 & 0 & 0 & a \\ 0 & 0 & -a & 0 \\ 0 & -a & 0 & -A \\ a & 0 & -A & 0 \end{array} \right)$
a and A are themselves 2x2 symmetric matrices: $a = \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{12} & a_{22} \end{array} \right)$ and $A = \left( \begin{array}{cc} A_{11} & A_{12} \\ A_{12} & A_{22} \end{array} \right)$.

Step 1: I diagonalize M treating a and A as numbers. Let U be an orthogonal matrix, then: $D = U^{-1} M U = \left( \begin{array}{cccc} -\frac{A}{2}-\frac{1}{2} \sqrt{A^2+4a^2} & 0 & 0 & 0 \\ 0 & -\frac{A}{2}+\frac{1}{2 }\sqrt{A^2+4a^2} & 0 & 0 \\ 0 & 0 & \frac{A}{2}-\frac{1}{2} \sqrt{A^2+4a^2} & 0 \\ 0 & 0 & 0 & \frac{A}{2}+\frac{1}{2} \sqrt{A^2+4a^2} \end{array} \right)$
Step 2: Under the assumption: $A>>a$ the square root becomes: $\sqrt{A^2+4a^2} ≈ A + 2\frac{a^2}{A}$ and consequently the four eigenvalues: $\pm A \pm \frac{a^2}{A}$ and $\pm \frac{a^2}{A}$. We can return now to our original a and A matrices and the diagonal matrix D becomes:
$D = \left( \begin{array}{cccc} -A-aA^{-1}a^{T} & 0 & 0 & 0 \\ 0 & aA^{-1}a^{T} & 0 & 0 \\ 0 & 0 & -aA^{-1}a^{T} & 0 \\ 0 & 0 & 0 & A+aA^{-1}a^{T} \end{array} \right)$
Step 3: If we now find an 8x8 matrix: $\left( \begin{array}{cccc} R_1 & 0 & 0 & 0 \\ 0 & R_2 & 0 & 0 \\ 0 & 0 & R_3 & 0 \\ 0 & 0 & 0 & R_4 \end{array} \right)$, where $R_i$ are the orthogonal matrices, which diagonalize the 2x2 eigenvalues of D, then we finally arrive at the matrix:
$D' = R^{-1} D R = diag(\lambda_i)$ with $\lambda_i$ the final eigenvalues with over 40 terms combining the various components of a and A.

Now I understand that probably nobody is gonna go through all this mess, but if some hero would do me this favour I have the following questions to ask:
• I want U to be a 4x4 orthogonal matrix parametrized with the four-dimensional Euler angles. Does somebody know how to find such a matrix? I know 4D rotations can be expressed with two quaternions and each quaternion corresponds to 3 angle parameteres. U would all in all be parametrized by 6 angles!
• $R_i$ on the other hand can be easily represented by a 2D rotation matrix. The difficult part is to find a closed expression for the one angle parameter as a function of the components of a and A. On condition $A>>a$, what further simplifications would you propose?

I bet this thread will remain barren, but what the heck :D

2. Jul 2, 2014