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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.

## What is Hermitian Matrix?

A square matrix is said to be a Hermitian matrix if it is equal to its conjugate transpose matrix. It is a square matrix that has complex numbers except for the diagonal entries, which are real numbers. We know that a complex number is a number that is expressed in the form of a + ib, where “a” is the real part and “b” is the imaginary part. The Hermitian matrix is named after the mathematician Charles Hermite.

A complex square matrix “An×n = [aij] is said to be a Hermitian matrix if

A = AH
where AH is the conjugate transpose of matrix A.

In other words, “An×n = [aij] is said to be a Hermitian matrix if aij= āji, where āji is the complex conjugate of aji.

### Examples of Hermitian Matrix

• Matrix given below is a Hermitian matrix of order “2 × 2.” Now, the conjugate of A ⇒ The conjugate transpose of matrix A ⇒ We can see that A = AH, so the given matrix is a Hermitian matrix.

• Matrix given below is a Hermitian matrix of order “3 × 3.” ## Properties of Hermitian Matrix

Some important properties of Hermitian matrix are discussed below:

• Principal diagonal entries of a Hermitian matrix are always real.
• Non-diagonal entries of a Hermitian matrix are complex numbers.
• If A is a Hermitian matrix of any order and k is a real scalar, then kA is also a Hermitian matrix as (kA)H = kAH = kA.
• When two Hermitian matrices of the same order are added or subtracted, the resulting matrix is also a Hermitian matrix.
• When two Hermitian matrices are multiplied, the resultant matrix is also a Hermitian matrix.
• Trace of a Hermitian matrix is always a real number.
• Determinant of a Hermitian matrix is always a real number.
• Inverse of the Hermitian matrix is also a Hermitian matrix.
• Conjugate matrix of a Hermitian matrix is also Hermitian.
• If A is a Hermitian matrix of any order, then An is also a Hermitian matrix for all positive integers n.

## Eigenvalues of Hermitian Matrix

Eigenvalues of a Hermitian matrix are always real.

For any Hermitian matrix A such that A’ = A and the eigenvalue of A be λ

Now, X is the corresponding Eigen vector such that AX = λX where,

X = Then X’ will be a conjugate row vector. Multiplying X, on both sides of AX = λX we have,

X’AX = X’λX = λ(X’X) = λ( a12 + b12 + ….. + an2 + bn2)

Here, ( a12 + b12 + ….. + an2 + bn2) is a real number

Now,

(X’AX)’ = X’A(X’)’ = X’AX,

hence X’AX is the Hermitian Matrix of order 1.

So X’AX is real, then, λ is also real.

## Skew-Hermitian Matrix

A complex square matrix is said to be a skew-Hermitian matrix if the conjugate transpose matrix is equal to the negative of the original matrix. A square matrix “An×n = [aij] is said to be a Hermitian matrix if AH = -A, where AH is the conjugate transpose of matrix A.

• Matrix given below is a Hermitian matrix of order “2 × 2.” Now, the conjugate of A ⇒ The conjugate transpose of matrix A ⇒ We can see that AH = −A, so the given matrix is a skew-Hermitian matrix.

## 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 = PH, 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 = MH, so the given matrix is a Hermitian matrix.

## FAQs on Hermitian Matrix

### Question 1: Define a matrix.

A rectangular array of numbers that are arranged in rows and columns is known as a “matrix.”

### Question 2: What is a Hermitian Matrix?

A square matrix is said to be a Hermitian matrix if it is equal to its conjugate transpose matrix. A complex square matrix “An×n = [aij] is said to be a Hermitian matrix if A = AH, where AH is the conjugate transpose of matrix A.

### Question 3: How to find whether a matrix is Hermitian or not?

Follow the steps given below to know whether the given matrix is Hermitian or not:

• First, find the conjugate matrix of the given matrix by replacing every element with its conjugate.
• Now, find the transpose of the resultant matrix.
• If the original matrix is equal to its conjugate transpose matrix, then the given matrix is Hermitian.