Skew Hermitian Matrix

If a square matrix A satisfies the condition A^H = - A, it is a skew Hermitian matrix. Here, A^H is the conjugate transpose of the square matrix A. To calculate A^H, we first take the transpose of the matrix A^T. After transposing, take the complex conjugate of each element in the matrix. This operation is denoted as A*.
For example, let a 2 × 2 matrix 
 

\( A = \begin{bmatrix} 0 & 2 + i \\ -2 + i & 0 \end{bmatrix} \)
 
To find the conjugate transpose A^H, we first transpose the matrix.
 

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

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


Now, we find -A.
 

\( - A = \begin{bmatrix} 0 & -2 - i \\ 2 - i & 0 \end{bmatrix} \)
  
As A^H = -A, so A is a skew Hermitian matrix.

Hermitian and skew Hermitian matrices can be differentiated based on their relations with the conjugate and transpose. Some common differences between the two are listed below.
 

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. a_ij = - a_ij​​—.
    
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 A^T
 

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

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.
 

For students, knowing how to identify and work with skew-Hermitian matrices helps when studying eigenvalues, matrix decompositions, and advanced topics in linear algebra. Here are some useful tips and tricks for effective learning: 

• Check the main condition first. Compute A^H and confirm if it equals −A.
   
• Look at diagonal entries. Ensure each diagonal element is either 0 or of the form 'bi' (purely imaginary).
   
• Use the relation between off-diagonals. For 𝑖 ≠ 𝑗, check whether a_ij=−aji^—.
   
• Connect to eigenvalues. Know that eigenvalues of a skew-Hermitian matrix are either 0 or purely imaginary.
   
• Relate to real case. If all entries are real, a skew-Hermitian matrix reduces to a skew-symmetric matrix. That is A^T = −A.

Skew Hermitian matrices are helpful in real-life situations involving complex systems. Some uses of these matrices across different fields are:

 

• Stability analysis in control systems: Skew Hermitian matrices have purely imaginary eigenvalues. This trait helps analyze the marginal stability of dynamic systems where the system oscillates stably. This property is used in designing stable feedback systems in robotics or aircraft systems.
  
   
• Complex signal filtering in signal processing: Skew Hermitian matrices model phase shifts and energy-preserving transformations in complex signal filters.
  
   
• Representing Impedance/Admittance in Electrical Engineering: These matrices are used in RLC networks and analyzing power flow in transmission lines as they represent some characteristics of impedance and admittance.
  
   
• Damped vibrational modes in engineering: Structural analysis in engineering and architecture requires skew Hermitian matrices to describe systems with damping and energy dissipation. 
  
   
• Lie algebra representation in theoretical Mathematics and Physics: Skew Hermitian matrices form the basis of Lie algebras used to study continuous symmetries. For example, fundamental concepts in quantum mechanics for modeling angular momentum and particle spin are based on skew Hermitian matrices.