Let $A$ and $B$ be two matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$ such that $$\forall ( X , Y ) \in \left( \mathcal { M } _ { n , 1 } ( \mathbb { R } ) \right) ^ { 2 } , \quad X ^ { \top } A Y = X ^ { \top } B Y .$$ Show that $A = B$.
Let $A$ and $B$ be two matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$ such that
$$\forall ( X , Y ) \in \left( \mathcal { M } _ { n , 1 } ( \mathbb { R } ) \right) ^ { 2 } , \quad X ^ { \top } A Y = X ^ { \top } B Y .$$
Show that $A = B$.