Hermitian and Skew-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.

The conjugate transpose of a matrix is found by changing the sign of every element's imaginary part and then taking the transpose of the matrix.

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

A = A^H

where,

• AH is the conjugate transpose of matrix A. 

In other words, "An × n = [a_ij] is said to be a Hermitian matrix if aij = ā_ji, where ā_ji is the complex conjugate of a_ji. The Hermitian matrix is named after the mathematician Charles Hermite. 

Examples of Hermitian Matrices

• The 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.

• The 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 a 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.
• The trace of a Hermitian matrix is always a real number.
• The determinant of a Hermitian matrix is always a real number.
• The inverse of the Hermitian matrix is also a Hermitian matrix.
• The 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 eigenvector 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 = [aij]" is said to be a Hermitian matrix if A^H = -A, where A^H is the conjugate transpose of matrix A.

The 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.

Properties of Skew-Hermitian Matrix

• Principal diagonal entries of a skew-Hermitian matrix are always purely imaginary (or zero).
• Non-diagonal entries are complex numbers such that a_{ij} = -\overline{a_{ji}} (negative of the complex conjugate of the symmetric position).
• If A is a skew-Hermitian matrix of any order and k is a real scalar, then kA is also a skew- Hermitian matrix because:
  (kA)^† = kA^† = −kA
• When two skew-Hermitian matrices of the same order are added or subtracted, the resulting matrix is also a skew-Hermitian matrix.
• When two skew-Hermitian matrices are multiplied, the product is a Hermitian matrix if and only if AB=BA (they commute).
• The trace of a skew-Hermitian matrix is purely imaginary (or zero).
• The determinant of a skew-Hermitian matrix is always a real number (and for odd order, it is a purely imaginary number times a real factor, often zero in the real case).
• The inverse of a skew-Hermitian matrix (if it exists) is also a skew-Hermitian matrix.
• The conjugate matrix of a skew-Hermitian matrix is skew-symmetric.
• If A is a skew-Hermitian matrix of any order, then Aⁿ is skew-Hermitian for odd n and Hermitian for even n.

Related Articles

• Minors and Cofactors of Determinants
• Determinant of a Matrix
• Adjoint of a Matrix

Solved Examples on Hermitian and Skew-Hermitian Matrices

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.

Example 5: Determine whether the matrix given below is skew-Hermitian or not:

M = \left[\begin{array}{ccc} 0 & i\\ -i & 0\\ \end{array}\right]

Solution:

Given matrix is

M = \left[\begin{array}{ccc} 0 & i\\ i & 0\\ \end{array}\right]

The conjugate transpose of matrix M ⇒ 

\overline{M} = \left[\begin{array}{ccc} 0 & -i\\ -i & 0\\ \end{array}\right]

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

Example 6: Determine whether the matrix given below is skew-Hermitian or not:

M = \left[\begin{array}{ccc} 5i & 2-3i & 4+i\\ -2-3i & 0 & 6i\\-4+i & 6i &-5i \\ \end{array}\right]

Solution:

Given matrix is

M = \left[\begin{array}{ccc} 5i & 2-3i & 4+i\\ -2-3i & 0 & 6i\\-4+i & 6i &-5i \\ \end{array}\right]

The conjugate of matrix M ⇒ 

\overline{M} = \left[\begin{array}{ccc} -5i & 2+3i & 4-i\\ -2+3i & 0 & -6i\\-4-i & -6i & 5i \\ \end{array}\right]

The conjugate transpose of matrix M ⇒ 

{M}^H = \left[\begin{array}{ccc} -5i & -2+3i & -4-i\\ 2+3i & 0 & -6i\\4-i & -6i & 5i \\ \end{array}\right]

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