Show that $\langle S,T\rangle = \operatorname{Tr}(ST)$ is an inner product on a vector space $V$ of of $3×3$ real symmetric matrices where $\operatorname{Tr}$ is trace of a matrix.
I've worked my way out to show that it is inner product.
But w.r.t. this inner product when I'm trying to find a matrix that is orthogonal to both $S_1 = \begin{pmatrix}1 & 0 & 1\\0 & 1 & 0\\1 & 0 &0\end{pmatrix}$ and $S_2 = \begin{pmatrix}1 & 0 & 1\\0 & -3 & 0\\1 & 0 &0\end{pmatrix}$, I can only find out using the definition if the inner product the elements $a,b,e$ of the orthogonal matrix $T =\begin{pmatrix}a & d & e\\d & b & f\\e & f &c\end{pmatrix}$.
How can I compute the other elements? Do i need to, because the entries that I found out are only determining the trace of their products to be 0?