Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$. I.B.1) Prove that the elements $a _ { i i } ( 1 \leqslant i \leqslant n )$ on the diagonal of $A$ are in $R ( A )$. I.B.2) By considering the matrix $$A = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$$ show that the elements $a _ { i j }$ with $i \neq j$ are not necessarily in $R ( A )$.
Let $A = \left( a _ { i j } \right) _ { 1 \leqslant i , j \leqslant n } \in \mathcal { M } _ { n } ( \mathbb { R } )$. We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
I.B.1) Prove that the elements $a _ { i i } ( 1 \leqslant i \leqslant n )$ on the diagonal of $A$ are in $R ( A )$.
I.B.2) By considering the matrix
$$A = \left( \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
show that the elements $a _ { i j }$ with $i \neq j$ are not necessarily in $R ( A )$.