Skew Hermitian Matrix
2026-02-28 01:26 Diff
https://brightchamps.com/en-us/math/algebra/skew-hermitian-matrix

Comparing A* and -A, we see that the condition A* = -A is satisfied.
So, A is a skew Hermitian matrix.


The elements of a skew Hermitian matrix follow these conditions:

  1. All diagonal elements are either purely imaginary or zero. 
  2. Non-diagonal elements can have both real and imaginary parts.


Based on these conditions, the general formula of a skew Hermitian matrix,
For a 2 × 2 skew Hermitian matrix is:
 

\( A = \begin{bmatrix} xi & y + zi \\ -y + zi & wi \end{bmatrix} \)

  For a 3 × 3 skew Hermitian matrix:

\( A = \begin{bmatrix} ai & b + ci & c + di \\ -b + ci & ei & g + hi \\ -c + di & -g + hi & ki \end{bmatrix} \)

What are the Properties of a Skew Hermitian Matrix?

Key properties of a skew Hermitian matrix include:
 

  1. All diagonal elements are purely imaginary or zero, i.e., of the form bi, where b R.
     
  2. The element at position (i, j) is the negative of the complex conjugate of the element at position (j, i) in a skew Hermitian matrix. aij = - aij​​—.
     
  3. The eigenvalues of a skew Hermitian matrix are either zero or purely imaginary.
     
  4. These matrices are always square.
     
  5. If all entries in a matrix are real numbers, then a skew Hermitian matrix becomes a skew-symmetric matrix. This is because taking the complex conjugate does not affect real numbers.
     
  6. The sum of two skew Hermitian matrices is also skew Hermitian.
     
  7. If A is skew Hermitian and α∈R then αA is also skew Hermitian.
     
  8. The zero matrix is a skew Hermitian matrix because 0* = -0 = 0.
     
  9. When a skew Hermitian matrix A, is multiplied by imaginary unit \(i = \sqrt-1\), we get the Hermitian matrix iA.

What is the condition for the Skew Hermitian Matrix?

For a matrix to be skew Hermitian, it must satisfy the condition A* = -A.
Let us take an example to check for the condition.
Let,
 

\( A = \begin{bmatrix} 0 & 3 + 2i \\ -3 + 2i & 0 \end{bmatrix} \)

  To check if A* = -A,
Find transpose AT

\( A^{T} = \begin{bmatrix} 0 & -3 + 2i \\ 3 + 2i & 0 \end{bmatrix} \)

   Complex conjugate A*

\( A^{*} = \begin{bmatrix} 0 & -3 - 2i \\ 3 - 2i & 0 \end{bmatrix} \)

   Now, we find -A

\( -A = \begin{bmatrix} 0 & -3 - 2i \\ 3 - 2i & 0 \end{bmatrix} \)
 

Skew Hermitian Matrix Eigenvalue

As established earlier, we see that the eigenvalues of a skew Hermitian matrix are purely imaginary or zero. So, to find these eigenvalues, we solve the characteristic equation \( \det(A - \lambda I) = 0 \). Here \(\lambda\) is an eigenvalue, and I is the identity matrix.