I have two generalized rayleigh quotient problems $$\max {R(F_1)}=\frac{(F_1^HAF_1)}{F_1^HBF_1}$$ $$ \max{R(F_2)}=\frac{(F_2^HAF_2)}{F_2^HCF_2}$$ where $A,B,C \in \Bbb C^{n\times n} $ are Hermitian matrices such that $$A={(x_1^Hx_1)},B={(y_1^Hy_1+aI)}, C={(x_1^Hx_1+y_1^Hy_1+aI)}$$ where $x_1 \Bbb C^ {l\times n},y_1 \in \Bbb C^{m\times n} $ such that $n>m>l$ and $x_1$ and $y_1$ are zero mean unit variance Gaussian distributed random variables

The maxima can be achieved by getting generalized eigen vectors for corresponding largest eigen values of $$AF_1=\lambda_{F1} BF_1$$ and $$AF_2=\lambda_{F2} CF_2$$ where $\lambda_{F1}$ and $\lambda_{F2}$ are ordinary eigen values of $B^{-1}A$ and $C^{-1}A$ It can be seen that maximum number of non zero eigen values are $l$ in both cases and we can get $l$ columns in eigen vectors $F_1$ and $F_2$.

My problem is there any relation between $F_1$ and $F_2$ due to additive relationship between $A,B$ and $C$ When I am simulating this in MATLAB I am getting exact same eigen vectors $F_1=F_2$ and when I am simulating this for $10^6$ different samples of $x_1,y_1$ they are giving similar soultion. How is it happening mathematically, can somebody help me to find a relation between $F_1$ and $F_2$


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