Symmetric matrix

This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices

Definition

Verbal definition

A symmetric matrix is a square matrix that equals its own matrix transpose. Explicitly, a square matrix $A$ is termed a symmetric matrix if $A=A^{T}$.

Algebraic definition

Suppose $n$ is a positive integer and $A=(a_{ij})_{1\leq i\leq n,1\leq j\leq n}$ is a $n\times n$ matrix. We say that $A$ is symmetric if the following holds:

$a_{ij}=a_{ji}\ \forall \ i\neq j\in \{1,2,\dots ,n\}$

Encoding

We can encode a symmetric matrix by simply storing the entries on or above the diagonal. This is a total of $n(n+1)/2$ entries instead of the total of $n^{2}$ entries.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
scalar matrix	scalar multiple of identity matrix			Bisymmetric matrix, Diagonal matrix|FULL LIST, MORE INFO
diagonal matrix	all off-diagonal entries are zero			|FULL LIST, MORE INFO
bisymmetric matrix	both symmetric and centrosymmetric			|FULL LIST, MORE INFO
symmetric Toeplitz matrix				Bisymmetric matrix|FULL LIST, MORE INFO

Other symmetry notions

• Persymmetric matrix
• Centrosymmetric matrix
• Bisymmetric matrix