Definition:Symmetric Matrix/Definition 1

Definition

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

$\mathbf A$ 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.

Also see

• Equivalence of Definitions of Symmetric Matrices

• Results about symmetric matrices can be found here.

Sources

• 1980: A.J.M. Spencer: Continuum Mechanics ... (previous) ... (next): $2.1$: Matrices: $(2.2)$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): symmetric matrix
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): symmetric matrix
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): symmetric matrix
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): symmetric matrix