issymmetric

Determine if a matrix is symmetric or skew-symmetric

Definitions

Symmetric Matrix

A symmetric matrix $M$ is a square matrix (real or complex) that satisfies the following condition:

For all $i$ and $j$, it holds that $m_{ij}=m_{ji}$, where $m_{ij}$ is the element in $i$-th row and $j$-th column of $M$.

Skew Symmetric Matrix

A skew symmetric matrix $M$ is a square matrix (real or complex) satisfying the following condition:

For all $i$ and $j$, it holds that $m_{ij}=-m_{ji}$, where $m_{ij}$ is the element in $i$-th row and $j$-th column of $M$.

This definition implies that all diagonal elements of $M$ are zero since $m_{kk}=-m_{kk}=0$ for all $k$.

Usage

issymmetric(A)

• A should be a numeric matrix. If A is a character array, it will be converted to a double array.
• If A is symmetric, it returns logical 1. Otherwise, it returns logical 0.
• If A is empty, it returns logical 1.
• If A is not square or ndims(A) > 2, it returns logical 0.

issymmetric(A, skew)

• A should be a numeric matrix. If A is a character array, it will be converted to a double array.
• skew has to be a character array, equal to either 'skew' or 'nonskew'.
• isymmetric(A,'nonskew') gives the same result as isymmetric(A).
• isymmetric(A,'skew') gives logical 1 when A is skew symmetric.
• If A is empty, it returns logical 1.
• If A is not square or ndims(A) > 2, it returns logical 0.

Examples

• B is a real symmetric matrix.

Input

A = [1 2 3;4 5 6;7 8 9];
B = A+A'
issymmetric(B)

Output

B =
2.000   6.000   10.00
6.000   10.00   14.00
10.00   14.00   18.00

ans =
1

• A is complex and skew-symmetric.

Input

A = [1 2 3;4 5 6;7 8 9]+[1 2 3;4 5 6;7 8 9] * i;
A(1,1) = 0; A(2,2) = 0; A(3,3) = 0;
A(1,2) = -A(2,1);
A(1,3) = -A(3,1);
A(2,3) = -A(3,2)
issymmetric(A,'skew')

Output

A =
0.000 + 0.000i  -4.000 - 4.000i  -7.000 - 7.000i
4.000 + 4.000i   0.000 + 0.000i  -8.000 - 8.000i
7.000 + 7.000i   8.000 + 8.000i   0.000 + 0.000i

ans =
1