How to Show a Matrix is Hermitian?

To demonstrate that a matrix is Hermitian, you need to verify that it's equal to its own conjugate transpose.

Here's a breakdown of the process:

1. Conjugate Transpose:

• Conjugate: Replace each element of the matrix with its complex conjugate. This means changing the sign of the imaginary part of each element.
• Transpose: Swap the rows and columns of the resulting matrix.

2. Comparison:

• Hermitian: If the original matrix and its conjugate transpose are identical, the matrix is Hermitian.

Example:

Let's consider the matrix:

A =  [ 2  1+i ]
     [ 1-i 3 ]

1. Conjugate:

A* = [ 2  1-i ]
     [ 1+i 3 ]

2. Transpose:

(A*)^T = [ 2  1+i ]
         [ 1-i 3 ]

3. Comparison:

Since A = (A*)^T, the matrix A is Hermitian.

Practical Insights:

• Hermitian matrices are essential in quantum mechanics and linear algebra.
• Their eigenvalues are always real, and their eigenvectors are orthogonal.
• Many physical quantities, like energy and momentum, are represented by Hermitian operators.