Sum of a matrix with its transpose

Question

I've a question that many of yours could consider stupid: if i sum a matrix with its transpose, I obtain a particular result? E.g. $A + A^T = B$, $B$ has some particular properties?

Answer 1

Yes, $B$ is symmetric: $B^T = B$.

Moreover, every symmetric $B$ can be written as $A+A^T$ for some matrix $A$. Indeed, take $A=\frac12 B$.