- #1
Trifis
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I am trying to diagonalize the following matrix:
M = <br /> \left( \begin{array}{cccc} <br /> 0 & 0 & 0 & a \\ <br /> 0 & 0 & -a & 0 \\<br /> 0 & -a & 0 & -A \\<br /> a & 0 & -A & 0 <br /> \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 =<br /> \left( \begin{array}{cccc} <br /> -\frac{A}{2}-\frac{1}{2} \sqrt{A^2+4a^2} & 0 & 0 & 0 \\ <br /> 0 & -\frac{A}{2}+\frac{1}{2 }\sqrt{A^2+4a^2} & 0 & 0 \\<br /> 0 & 0 & \frac{A}{2}-\frac{1}{2} \sqrt{A^2+4a^2} & 0 \\<br /> 0 & 0 & 0 & \frac{A}{2}+\frac{1}{2} \sqrt{A^2+4a^2} <br /> \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 =<br /> \left( \begin{array}{cccc} <br /> -A-aA^{-1}a^{T} & 0 & 0 & 0 \\ <br /> 0 & aA^{-1}a^{T} & 0 & 0 \\<br /> 0 & 0 & -aA^{-1}a^{T} & 0 \\<br /> 0 & 0 & 0 & A+aA^{-1}a^{T}<br /> \end{array} \right)
Step 3: If we now find an 8x8 matrix: \left( \begin{array}{cccc} <br /> R_1 & 0 & 0 & 0 \\ <br /> 0 & R_2 & 0 & 0 \\<br /> 0 & 0 & R_3 & 0 \\<br /> 0 & 0 & 0 & R_4<br /> \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 going to go through all this mess, but if some hero would do me this favour I have the following questions to ask:
I bet this thread will remain barren, but what the heck :D
M = <br /> \left( \begin{array}{cccc} <br /> 0 & 0 & 0 & a \\ <br /> 0 & 0 & -a & 0 \\<br /> 0 & -a & 0 & -A \\<br /> a & 0 & -A & 0 <br /> \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 =<br /> \left( \begin{array}{cccc} <br /> -\frac{A}{2}-\frac{1}{2} \sqrt{A^2+4a^2} & 0 & 0 & 0 \\ <br /> 0 & -\frac{A}{2}+\frac{1}{2 }\sqrt{A^2+4a^2} & 0 & 0 \\<br /> 0 & 0 & \frac{A}{2}-\frac{1}{2} \sqrt{A^2+4a^2} & 0 \\<br /> 0 & 0 & 0 & \frac{A}{2}+\frac{1}{2} \sqrt{A^2+4a^2} <br /> \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 =<br /> \left( \begin{array}{cccc} <br /> -A-aA^{-1}a^{T} & 0 & 0 & 0 \\ <br /> 0 & aA^{-1}a^{T} & 0 & 0 \\<br /> 0 & 0 & -aA^{-1}a^{T} & 0 \\<br /> 0 & 0 & 0 & A+aA^{-1}a^{T}<br /> \end{array} \right)
Step 3: If we now find an 8x8 matrix: \left( \begin{array}{cccc} <br /> R_1 & 0 & 0 & 0 \\ <br /> 0 & R_2 & 0 & 0 \\<br /> 0 & 0 & R_3 & 0 \\<br /> 0 & 0 & 0 & R_4<br /> \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 going to 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