Let $A,B$ be matrices and $c,d,X$ be vectors... Can I find a vector $e$ and a Matrix $C$ such that $$ A(X-c) + c + B(X-d) +d = C(X-e) +e? $$ It seems so simple but I cannot solve it... I've tried a few tricks and the furthest I have seem to have gotten to is to reduce the problem to solving for: $$ (A+B)(X- (c+d)) +Ad +Bc +c +d ... $$ However this does not seem to lead anywhere but down the rabbithole..
2026-03-25 12:20:51.1774441251
Wierd Factorization of Matrix polynomial
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Rewrite the equation as $(C-I)(X-e)=v:=A(X-c) + c + B(X-d) +d-X$. You are essentially asking that given a vector $v$, whether there exists a matrix $M\,(=C-I)$ and a vector $x\,(=X-e)$ such that $Mx=v$. Clearly, you may just pick $x=(1,0,\ldots,0)^T$ and any matrix $M$ whose first column is $v$.