Definition:Symmetric Matrix

Definition

Let $\mathbf A$ be a square matrix over a set $S$.

Definition $1$

$\mathbf A$ is a symmetric matrix if and only if:

$\mathbf A = \mathbf A^\intercal$

where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.

That is, if and only if:

$a_{i j} = a_{j i}$

for all $i$ and $j$ for which $a_{i j}$ is defined.

Definition $2$

$\mathbf A$ is a symmetric matrix if and only if it is symmetrical about its main diagonal.

Also see

• Equivalence of Definitions of Symmetric Matrices

• Results about symmetric matrices can be found here.