Solved Examples on Hermitian Matrix

Example 1: Determine whether the matrix given below is a Hermitian matrix or not.

Solution:

Given matrix is 

Now, the conjugate of P ⇒ 

The conjugate transpose of matrix P ⇒ 

We can see that P = P^H, so the given matrix is a Hermitian matrix.

Example 2: Prove that the trace of a Hermitian matrix is always a real number.

Solution:

Let us consider a “2 × 2” Hermitian matrix to prove that its trace is always a real number.

Here, a, b, c, and d are real numbers.

We know that the trace of a matrix is the sum of its principal diagonal entries.

So, the trace of the matrix Q = a + d

As a and d are real numbers, a + d is also real.

So, the trace of the given Hermitian matrix is a real number.

Similarly, we can consider any Hermitian matrix of any other order and check that its trace is a real number.

Hence proved.

Example 3: Prove that the determinant of a Hermitian matrix is always real.

Solution:

Let us consider a “2 × 2” Hermitian matrix to prove that its determinant is always a real number.

Here, a, b, c, and d are real numbers.

det A = ad − (b + ci) (b−ci)

|A| = ad − [b² − c²i²]

|A| = ad − [b² − c² (−1)]

|A| = ad −b² − c² = real number

So, the determinant of the given Hermitian matrix is a real number.

Similarly, we can consider any Hermitian matrix of any other order and check that its determinant is a real number.

Hence proved.

Example 4: Determine whether the matrix given below is a Hermitian matrix or not.

Solution:

Given matrix is

The conjugate transpose of matrix M ⇒ 

The conjugate transpose of matrix M ⇒ 

We can see that M = M^H, so the given matrix is a Hermitian matrix.

A rectangular array of numbers that are arranged in rows and columns is known as a “matrix.” The size of a matrix can be determined by the number of rows and columns in it. If a matrix has “m” rows and “n” columns, then it is said to be an “m by n” matrix and is written as an “m × n” matrix. For example, a matrix with five rows and three columns is a “5 × 3” matrix. We have various types of matrices, like rectangular, square, triangular, symmetric, singular, etc. Now let us discuss the Hermitian matrix in detail.