ishermitian

Determine if a matrix is Hermitian or skew-Hermitian

Definitions

Hermitian Matrix

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

If $m_{rc}=a+bi$ is the element in Row $r$ and Column $c$ of $M$, then $m_{cr}=a-bi$ for all $r$ and $c$. That means, $m_{cr}$ is the conjugate of $m_{rc}$.

• This condition implies that the diagonal element $m_{kk}=a$ must be real for all $k$.
• When $M$ is real, it is also called symmetric.

Skew Hermitian Matrix

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

If $m_{rc}=a+bi$ is the element in Row $r$ and Column $c$ of $M$, then $m_{cr}=-a+bi$ for all $r$ and $c$. That means, $m_{cr}$ is the negation of the conjugate of $m_{rc}$.

• This condition implies that the diagonal element $m_{kk}=bi$ must be purely imaginary for all $k$.
• When $M$ is real, it is called skew symmetric.

Usage

ishermitian(A)

• A should be a numeric matrix. If A is a character array, it will be converted to a double array.
• If A is hermitian, 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.

ishermitian(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'.
• ishermitian(A,'nonskew') gives the same result as ishermitian(A).
• ishermitian(A,'skew') gives logical 1 when A is skew Hermitian.
• 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 Hermitian matrix.

Input

A = [1 2 3;4 5 6;7 8 9];
B = A+A'
ishermitian(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 skew Hermitian.

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) = -conj(A(2,1));
A(1,3) = -conj(A(3,1));
A(2,3) = -conj(A(3,2))
ishermitian(A,'skew')

Output

A = 
 0.0000 + 0.0000i  -4.0000 + 4.0000i  -7.0000 + 7.0000i
 4.0000 + 4.0000i   0.0000 + 0.0000i  -8.0000 + 8.0000i
 7.0000 + 7.0000i   8.0000 + 8.0000i   0.0000 + 0.0000i

ans = 
1