Hermitian Matrix

A Hermitian matrix is a square matrix over the complex numbers that equals its conjugate transpose, that is, AH = A. The conjugate transpose of matrix A, which is AH, can be found by first switching its rows and columns to get the transpose, and then taking the complex conjugate of each entry. This means a Hermitian matrix stays the same after taking its conjugate transpose. Hermitian matrices can be of any square size, such as 2×2, 3×3, 4×4, and so on.
 

A Hermitian matrix of order 2 × 2 is a square matrix whose entries satisfy the condition A = AH, where AH is the conjugate transpose. In the matrix, the element in the first row and second column is the complex conjugate of the element in the second row and first column, and the diagonal elements are real numbers in the matrix. 

A 2 × 2 Hermitian matrix looks like:

1       2-7i

2 + 7i     3

A Hermitian matrix of order 3 × 3 is a square matrix with complex numbers such that it is equal to its conjugate transpose. For example, 

4      ​2+i    3​      
2−i   5−i    3 
i        7      6

In the Hermitian matrix, all diagonal elements must be real numbers. Off-diagonal elements must be complex conjugates of each other across the main diagonal.
 

The Hermitian matrix has several properties that help improve our understanding of these matrices.

• In a Hermitian matrix, all the diagonal elements are real numbers. 
  
   
• Off-diagonal elements must be complex conjugates of each other across the main diagonal.
  
   
• Every Hermitian matrix is a normal matrix because it satisfies the condition 
  
   
• AAH = AHA. In fact, for Hermitian matrices, A = AH.
  
   
• Adding two Hermitian matrices results in another Hermitian matrix.
  
   
• If the Hermitian matrix is invertible, then its inverse is also Hermitian.
   

The eigenvalues of a Hermitian matrix are always real, even if the matrix itself contains complex numbers. If a matrix multiplies a vector and the result is just a scaled version of that same vector, the scaling factor is called an eigenvalue, which is denoted by 𝜆(lambda). An eigenvalue λ of a matrix A is a scalar such that AX = λX, where X is a non-zero eigenvector.

Skew-Hermitian Matrix

A skew-Hermitian matrix is a square matrix that is equal to the negative of its own conjugate transpose. It is written as A* = AT, where A* is the conjugate transpose of A. For example, let’s take a 2×2 matrix into consideration:

0    3 -i
-3 -i   0

A = 

Flip the matrix across its diagonal and take the complex conjugate of all its elements.

0   -3+i
3 +i   0

                               A* = -A

Like all other matrices, the Hermitian matrix also has a wide range of real-life applications. Some of them are discussed below:

Physics: A Hermitian matrix represents measurable quantities like energy, momentum, and spin in quantum physics. Since the eigenvalues of a Hermitian matrix are always real, they ensure that the outcomes of measurements (like an electron’s energy) are also real, which is physically meaningful.

Machine Learning & Data Science: Hermitian matrices are used to analyze complex-valued data. Since the covariance matrix is Hermitian, it guarantees real eigenvalues, which makes interpreting the data and reducing its dimensions more reliable.

Image Compression & Processing: In image processing, a Hermitian matrix is used in image transformation when applying the Discrete Fourier Transform (DFT) on images. Because the DFT of real-valued data exhibits Hermitian symmetry, this property helps reduce storage and improve compression methods like JPEG and IMG.

Finance (Risk Modeling): Hermitian or symmetric matrices are used in finance to represent the correlation between assets. They ensure that calculations like portfolio variance remain valid with real and interpretable results.

Telecommunications: Hermitian matrices play a role in radar and wireless systems, especially in techniques like beam forming. They help determine the best way to direct signals from multiple antennas, improving signal clarity and reducing interference.