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I am examining when the following expression is true:

$$tr( \sum_{i\in G_1}^{k} {A_iB_iC_iB_i}) - tr(\sum_{i\in G_2}^{l} {A_jB_jC_jB_j}) = 0 \quad \quad \quad \: - (*)$$

where

$A_{i},B_{i},C_{i}\quad (\forall i \in G_1 )$ - symmetric and positive definite matrices of the dimension $ n\times n $. $A_{j},B_{j},C_{j}\quad(\forall j \in G_2 )$ - symmetric and positive definite matrices of the dimension $ n\times n $ $G_1 = 1,2,..,k$ ; $G_2 = 1,2,..,l$ ;

Numerically this will require $o(n^2)$ to solve. However, I am trying to evaluate the cases (in general) where this goes to zero.

Note: I am not considering the trivial solution (i.e. when $l=k$ and $A_{i}=A_{j},B_{i}=B_{j},C_{i}=C{j}\: \forall i,j= 1,2,..l $ )

What have I tried?

I have tried to do this for two-dimensional matrices and it gets messy! I also couldn't find (I am still trying) trace/multiplication properties that would simplify the equation! I could put the sum inside the trace but That does not make the problem easier.

Additional problem background information

This is an equation that I got after solving the hamiltonian and now I am trying to find the singular arch for a costate. In (*), $A$ is another costate that can be assumed to be positive definite all the time. $B$ is the estimate covariance error of some variables. $C = H^T R^{-1} H $; where $R$ is positive definite ($R$ is al)

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