Consider V the space of n x n matrices, and S the subspace of simmetrical n x n matrices. Consider de dot product between A and B defined by the sum of the elements of the diagonal of AB^T.
Prove that P(A)=(A+A^T)/2 is the projection of an element A of V onto S.
Can anyone solve this?

Question

Consider V the space of n x n matrices, and S the subspace of simmetrical n x n matrices. Consider de dot product between A and B defined by the sum of the elements of the diagonal of AB^T.
Prove that P(A)=(A+A^T)/2 is the projection of an element A of V onto S.
Can anyone solve this?

helder_edwin · Answer

so u have
$$\large V=\mathbb{R}^{n\times n}\qquad V\geq S=\{A\in V:A^t=A\} $$
and the inner product
$$\large (A|B)=\text{tr}(AB^t)=\sum_{k=1}^n(AB^t)_{kk}=
\sum_{k=1}^n\sum_{h=1}^nA_{kh}B^t_{hk} $$