Hermitian Matrix

Hermitian Matrix is a special matrix; etymologically, it was named after a French Mathematician Charles Hermite (1822 – 1901), who was trying to study the matrices that always have real Eigenvalues. The Hermitian matrix is pretty much comparable to a symmetric matrix. The symmetric matrix is equal to its transpose, whereas the Hermitian matrix is equal to its conjugate transpose, sometimes referred to as tranjugate.

The Hermitian matrix has complex numbers; however, its diagonal entries are real. The Eigenvalues of a Hermitian matrix are always real.

Let us learn more about Hermitian matrices and their properties in detail, along with hermitian matrix examples.

What is a Hermitian Matrix?

A Hermitian matrix is a matrix that is equal to its conjugate transpose. Mathematically, a Hermitian matrix is defined as

For example, let

Then conjugate of A =

and transpose of conjugate of A =

Therefore, A* = A

Hence, A is a Hermitian Matrix.

Properties of a Hermitian Matrix

Let us take a note of some important properties of a Hermitian matrix.

• A square matrix A of order n is Hermitian if and only if every a_ij ∈ A,
  \(\begin{array}{l}\overline{a_{ij}}=a_{ij}\end{array} \)
  (1≤ i, j ≤ n).
• If A is a Hermitian matrix, and k is any real scalar, then kA is also a Hermitian matrix.

Let us take (kA)* = kA* = kA {since k is a real number}

• If A is a Hermitian matrix, the (A*)* = A

As (A*)* = (A)* = A* =A.

• If A and B are two additional conformable Hermitian matrices, then A + B is also Hermitian.

Given A and B are Hermitian, then

=

⇒ (A + B)* = A + B is Hermitian.

• If A and B are square matrices, then (AB)* = B*A*. If A and B are Hermitian, then (AB)* = BA.
• The determinant of a Hermitian matrix is real.
• The inverse of a Hermitian matrix is Hermitian as well.
• Conjugate of a Hermitian matrix is also Hermitian.
• If A is Hermitian, then A*A and AA* is also Hermitian.
• Any square matrix can be represented as A + iB, where A and B are Hermitian matrices.
• For any square matrix A, if A* = – A, then A is called the skew-Hermitian matrix. Any square matrix can be uniquely represented as a sum of a Hermitian and skew-Hermitian matrix.

Let A be any square matrix, then

A = ½(A + A*) + ½(A – A*); where (A + A*) is Hermitian and (A – A)* is skew-Hermitian.

• If A is Hermitian matrix, then Aⁿ is also Hermitian for all positive integers n.

Given A is Hermitian, that is, A* = A

Now, (Aⁿ)* = (A*)ⁿ = Aⁿ

⇒ (Aⁿ)* = Aⁿ is Hermitian.

• Trace of a Hermitian matrix is always real.

Eigenvalues of Hermitian Matrix

The Eigenvalues of a Hermitian matrix are always real.

Let A be a Hermitian matrix such that A* = A and λ be the eigenvalue of A.

Let X be the corresponding Eigen vector such that AX = λX where

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

X*AX = X*λX = λ(X*X) = λ( a₁² + b₁² + ….. + a_n² + b_n²)

Clearly, ( a₁² + b₁² + ….. + a_n² + b_n²) is real.

Now, (X*AX)* = X*A(X*)* = X*AX, hence X*AX is Hermitian of order 1.

Thus X*AX is a real number, consequently λ is also real.

Let us take an example,

Clearly, A* = A, A is Hermitian.

The characteristic polynomial of A is

|A – λI| =

= ( 3 – λ)( 2 – λ) – [(1 – i)(1 + i)]

= λ² – 5λ + 4

= (λ – 1)(λ – 4)

Thus, the eigenvalues of A are 1 and 4 which are real.

Related Articles

• Types of Matrices
• Transpose of Matrix
• Symmetric and Skew-Symmetric Matrix
• Eigenvalue of a Matrix
• Unitary Matrix

Solved Examples on Hermitian Matrix

Example 1:

Check whether the given matrix

Solution:

Given,

Conjugate of A =

Conjugate transpose of A = A* =

Hence A* = A

Thus, A is Hermitian.

Example 2:

If k is complex number and A be a Hermitian matrix. Will kA be Hermitian?

Solution:

Given A is a Hermitian matrix, A* = A and k is any complex number

Now,

Since k is a complex number, and it is not equal to its conjugate.

Hence, (kA)* ≠ kA

Thus, kA is not a Hermitian matrix.