Adding matrices to their transpose

Question

If I add a matrix to its transpose matrix, will I get a symmetric matrix? Is $B := A + A^\top$ symmetric?

Answer 1

Yes

$$B^T=(A+A^T)^T=A^T+(A^T)^T=A^T+A=A+A^T=B$$

Note $A$ and $A^T$ must have the same dimensions, hence $A$ must be square.

Answer 2

Yes (when $A$ is a square matrix of dimension $n\times n$).

Let $1\le i,j,k\le n$.

• The diagonal elements of both $A$ and $A^T$ are the same ($\{a_{ii}\}$) so that of $A+A^T$ are $2\times\{a_{ii}\}$.

• Also, a non-diagonal element of $A$ has the form $\{a_{jk}\}$ where $j\neq k$ and that of $A^T$ has the form $\{a_{kj}\}$. So the element of $A+A^T$ on the $j$th row and $k$th column is the same as that on the $k$th row and $j$th column - both are equal to $a_{jk}+a_{kj}=a_{kj}+a_{jk}$.

Combining these two yields the result.