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.

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, if and only if AB = BA.

• 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^H is 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

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.