Symmetric and Skew Symmetric Matrices

A square matrix $A$ is called symmetric when it is unchanged by transposition, that is $A' = A$ — every element mirrored across the main diagonal is identical to the entry on the other side, $a_{ij} = a_{ji}$ for all $i, j$. A square matrix is called skew-symmetric when transposing it flips the sign of every entry instead: $A' = -A$, which forces every diagonal element to be zero (since $a_{ii} = -a_{ii}$ only when $a_{ii} = 0$).