Symmetry

Definition

A matrix $M$ is symmetric if it equals its own transpose:

$$M=M^T$$

For a 2x2 matrix, this means that the two off-diagonal elements must be equal:

$$M=\left(\begin{array}{c}ab\\ bc\end{array}\right)$$

Antisymmetry

Definition

A matrix $M$ is antisymmetric if it equals the negative of its own transpose:

$$M=-M^T$$

For a 2x2 matrix, this means that all diagonal elements must be equal to $0$, since $a=-a$ implies $a=0$, and that the off-diagonals are negatives of each other:

M = \begin{pmatrix} 0 \quad b \\ -b \quad 0 \end{pmatrix} $$0 \quad b \\ -b \quad 0