Let $V$ be the real vector space of $3 \times 3$ matrices with the bilinear form $\langle A,B \rangle=$ trace $A^tB$, and let $W$ be the subspace of skew-symmetric matrices. Compute the orthogonal projection to $W$ with respect to this form, of the matrix $$\begin{pmatrix} 1& 2 & 0\\ 0 & 0 & 1\\ 1 & 3 & 0\end{pmatrix}$$
Could someone show me how to proceed ?
Best Answer
Hint If $A$ is symmetric and $B$ is skew-symmetric, then $$ \mathrm{tr} (A^TB)=\mathrm{tr} (A^TB)^T=\mathrm{tr} (B^TA)=\mathrm{tr} (AB^T)=-\mathrm{tr} (A^TB)\quad \Rightarrow \quad \mathrm{tr} (A^TB)=0 $$
Hence
Therefore