Let $a,a',a'',b,b',b'',x$ be square matrices with real entries.
Notice matrix multiplication is not commutative , but matrix miltiplication behaves as a noncommutative ring.
I know we can " slightly simplify " $ A X + X B $ to $( A + Q) X$ where $ Q X = X B $. ( or similar to the other side by replacing $A$ INSTEAD of $B$ )
However does this mean we Cannot simplify the expression
$$a x b + a' x b' + a'' x b'' ? $$
Neither theoretically nor algorithmic (*) ? ( * assuming we already have the optimal algoritm for matrix addition and also for matrix multiplication. )
Distributive property , diagonalize and matrix inverses seem not to help here , if i am correct.
Im not an expert in linear algebra or matrices.