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:

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

Note that the $i\neq j$ condition is not required; the above definition is equivalent to:

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

Finally, because the equality condition is symmetric in $i,j$ , it suffices to check it for $i<j$ , so the above definition if equivalent to:

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

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.

Small cases

For the equal entries $a_{ij}$ and $a_{ji}$ , $i<j$ , we will use $a_{ij}$ as the label. This is purely a matter of convention.

$n$	General description of $n\times n$ symmetric matrix	Number of free parameters (equals $n(n+1)/2$ )
1	${\begin{pmatrix}a_{11}\\\end{pmatrix}}$	1
2	${\begin{pmatrix}a_{11}&a_{12}\\a_{12}&a_{22}\\\end{pmatrix}}$	3
3	${\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{12}&a_{22}&a_{23}\\a_{13}&a_{23}&a_{33}\\\end{pmatrix}}$	6

Matrix operations

Invariant computation

For both the trace and the determinant, there does not seem to be a simpler way of computing them than the standard approach for square matrices.

Operations

Matrix operation	Description in terms of encoding	Number of additions	Number of multiplications	Number of inverse computations	Set of symmetric matrices closed under operation?
Addition of two symmetric matrices	Add the corresponding entries of the two matrices	$n(n+1)/2$	0	0	Yes
Multiplication of two symmetric matrices	Usual matrix multiplication algorithms; symmetry does not seem to offer speedup	$n^{2}(n-1)$ for naive matrix multiplication	$n^{3}$ for naive matrix multiplication	0	No

Relation with other properties

Stronger properties

Property	Meaning	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