Suppose you have an equation ${{d\bf{A}}\over{dx}}= A B^\dagger A - B$,
where A,B are $n\times n$ unitary matrices ;
B is defined by $B_{ij}= -\frac{1}{2}\alpha_i\delta_{ij}A_{ij}$.
The expression isn't particularly important but note that the RHS is pretty complicated - B being a projection of A of sorts.
The fixed points of this equation are determined by equating the RHS to a zero matrix. I need to -
- find out the fixed points to this system; for the $3\times 3$ case all of these may be guessed, except for one.
- calculate the stabilities around these fixed points.
EDIT: by "calculating stabilities about a fixed point" i mean the following -
(i) Obtain the fixed point(s) $A^*$ such that ${{d\bf{A}}\over{dx}}\vert_{A=A^*} = f(A^*)= 0$
(ii) if $A^*\mapsto A^*+\epsilon A_1$, the RHS changes, upto the first order in $\epsilon$ to a slightly larger function easily got at by the product rule of differentiation (note that the differential of $B$ is still the projection of $A_1$). Now one has to diagonalise this expression, to obtain choices of $A_1$ that satisfy the form $ {{d\epsilon}\over{dx}} = \mu \epsilon \vert_{\mbox{along $A_1$}} $ Note that any unitary transformation on A renders the concise relation between A and B moot.
Please suggest analytical approaches to tackle this problem.
EDIT: The fixed points of the equation ${{d\bf{A}}\over{dx}}= A B^\dagger A - A$ with the $\alpha_i$s set to one - are -
(i) The identity element of GL3 - $\left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$ (ii-iv) Cyclic Permutations of - $\left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right)$ (v-vi) And the two cases - $\left(\begin{array}{rrr} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)$ and $\left(\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right)$
(vii) Define $a\equiv \frac{1}{\sum_i \alpha_i^{-1}}$. The fixed point is - $ A_{ij} = -\frac{a}{\alpha_i}\delta_{ij} + \sqrt{( 1 - \frac{a}{\alpha_i})( 1 - \frac{a}{\alpha_j})}(1-\delta_{ij}) $ Hope this helps! Code for the matrices are here.
If you think you can help with a numerical method for this, visit here.