Hermitian Matrix

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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 “A_n×n = [a_ij] is said to be a Hermitian matrix if

A = A^H
where A^H is the conjugate transpose of matrix A.

In other words, “A_n×n = [a_ij] is said to be a Hermitian matrix if a_ij= ā_ji, where ā_ji is the complex conjugate of a_ji.

Examples of Hermitian Matrix

Matrix given below is a Hermitian matrix of order “2 × 2.”

$A = \left[\begin{array}{cc} 8 & 1+i\\ 1-i & 5 \end{array}\right]$

Now, the conjugate of A ⇒

$\bar{A}= \left[\begin{array}{cc} 8 & 1-i\\ 1+i & 5 \end{array}\right]$

The conjugate transpose of matrix A ⇒

$A^{H} = (\bar{A})^{T} \left[\begin{array}{cc} 8 & 1+i\\ 1-i & 5 \end{array}\right]$

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

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

$B = \left[\begin{array}{ccc} 1 & 2+3i & 4i\\ 2-3i & 0 & 6-7i\\ -4i & 6+7i & 3 \end{array}\right]$

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 = kA^H = 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 = $\begin{bmatrix} a_{1}+ib{_{1}} \\ a_{2}+ib{_{2}} \\ ...\\ ...\\ a_{n}+ib{_{n}} \\ \end{bmatrix}$

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 + ….. + a_n² + b_n²)

Here, ( a₁² + b₁² + ….. + a_n² + b_n²) 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 “A_n×n = [a_ij] is said to be a Hermitian matrix if A^H = -A, where A^His the conjugate transpose of matrix A.

Matrix given below is a Hermitian matrix of order “2 × 2.”

$A = \left[\begin{array}{cc} ai & -b+ci\\ -b-ci & 0 \end{array}\right]$

Now, the conjugate of A ⇒

$\overline{A} = \left[\begin{array}{cc} -ai & -b-ci\\ -b+ci & 0 \end{array}\right]$

The conjugate transpose of matrix A ⇒

$A^{H} = (\overline{A})^{T} = \left[\begin{array}{cc} -ai & -b+ci\\ -b-ci & 0 \end{array}\right] = -A$

We can see that A^H = −A, so the given matrix is a skew-Hermitian matrix.

Read More,

Minors and Cofactors of Determinants

Determinant of a Matrix

Adjoint of a Matrix

Solved Examples on Hermitian Matrix

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

$P = \left[\begin{array}{cc} -7 & 2+5i\\ 2-5i & 3 \end{array}\right]$

Solution:

Given matrix is $P = \left[\begin{array}{cc} -7 & 2+5i\\ 2-5i & 3 \end{array}\right]$

Now, the conjugate of P ⇒ $\bar{P} = \left[\begin{array}{cc} -7 & 2-5i\\ 2+5i & 3 \end{array}\right]$

The conjugate transpose of matrix P ⇒

$P^{H} = (\bar{P})^{T}= \left[\begin{array}{cc} -7 & 2+5i\\ 2-5i & 3 \end{array}\right] = P$

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.

$A = \left[\begin{array}{cc} a & b+ci\\ b-ci & d \end{array}\right]$

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.

$A = \left[\begin{array}{cc} a & b+ci\\ b-ci & d \end{array}\right]$

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.

$M = \left[\begin{array}{ccc} 0 & 5+7i & 3i\\ 5-7i & 9 & 1-2i\\ -3i & 1+2i & -11 \end{array}\right]$

Solution:

Given matrix is

$M = \left[\begin{array}{ccc} 0 & 5+7i & 3i\\ 5-7i & 9 & 1-2i\\ -3i & 1+2i & -11 \end{array}\right]$

The conjugate transpose of matrix M ⇒

$\overline{M} = \left[\begin{array}{ccc} 0 & 5-7i & -3i\\ 5+7i & 9 & 1+2i\\ 3i & 1-2i & -11 \end{array}\right]$

The conjugate transpose of matrix M ⇒

$M^{H} = (\overline{M})^{T} = \left[\begin{array}{ccc} 0 & 5+7i & 3i\\ 5-7i & 9 & 1-2i\\ -3i & 1+2i & -11 \end{array}\right] = M$

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

FAQs on Hermitian Matrix

Question 1: Define a matrix.

Answer:

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

Question 2: What is a Hermitian Matrix?

Answer:

A square matrix is said to be a Hermitian matrix if it is equal to its conjugate transpose matrix. A complex square matrix “A_n×n = [a_ij] is said to be a Hermitian matrix if A = A^H, where A^H is the conjugate transpose of matrix A.

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

Answer:

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.

Question 4: What is a skew-Hermitian Matrix?

Answer:

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.

Question 5: What is the nature of the determinant of a Hermitian matrix?

Answer:

The determinant of a Hermitian matrix is always a real number.

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Last Updated : 26 Mar, 2023

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