Skew Hermitian Matrix

A skew-Hermitian matrix is the anti of a Hermitian matrix which is why the skew-Hermitian matrix is also known as the anti-Hermitian matrix. The skew-Hermitian matrix is closely similar to that of a skew-symmetric matrix. A skew-symmetric matrix is equal to the negative of its transpose; similarly, a skew-Hermitian matrix is equal to the negative of its conjugate transpose.

Thus, a skew-Hermitian matrix satisfies the properties opposite to that of a Hermitian matrix, which was named after a French mathematician Charles Hermite. He came across the concept of these types of matrices while studying for the matrix, which always has real eigenvalues.

Table of Contents:

What is a Skew Hermitian Matrix?

A square matrix of complex numbers is said to be a skew-Hermitian matrix if that matrix is equal to the negative of its conjugate transpose, also called tranjugate. Mathematically, the definition of the skew-Hermitian matrix is as follows:

A square matrix A = [aij] n × n is such that A* = −A, that is, for every aij ∈ A,
\(\begin{array}{l}\overline{a_{ij}} = -a_{ij}\end{array} \)
(1≤ i, j ≤ n), then A is called a skew-Hermitian matrix.

Where

\(\begin{array}{l}A^{*} = \overline{A^{T}}; \text{ conjugate transpose of A. }\end{array} \)

Thus, A = [aij] n × n is a skew-Hermitian matrix ⇔ A* = −A

Example of a Skew Hermitian Matrix

Let us understand more about skew-Hermitian matrix by taking an example.

\(\begin{array}{l}\text{Let A be a square matrix such that  }A = \begin{bmatrix} 0 & 1+2i & 3+i \\ -1+2i & 2i & 1-i \\ -3+i & -1-i & -3i \\\end{bmatrix}\end{array} \)
\(\begin{array}{l}\text{Now the conjugate of A =  }\bar{A} = \begin{bmatrix} 0 & 1-2i & 3-i \\ -1-2i & -2i & 1+i \\ -3-i & -1+i & 3i \\\end{bmatrix}\end{array} \)

\(\begin{array}{l}A^{*}=\begin{bmatrix}0 & -1-2i & -3-i \\1-2i & -2i & -1+i \\3-i & 1+i & 3i \\\end{bmatrix}=-\begin{bmatrix}0 & 1+2i & 3+i \\-1+2i & 2i & 1-i \\-3+i & -1-i & -3i \\\end{bmatrix}= -A\end{array} \)

Hence, A is a skew-Hermitian matrix.

Formula of Skew Hermitian Matrix

For a skew-Hermitian matrix of any order, we may find some interesting facts:

  • The diagonal entries are either purely imaginary or zero.
  • Elements other than diagonal elements may have real as well as imaginary parts.
  • The imaginary part of the ith row and jth column, other than diagonal elements, is the same.
  • The real part of the ith row and jth column, other than diagonal elements, is the same but have opposite signs.

With the above observations, we have the general form of a skew-Hermitian matrix is:

\begin{bmatrix} ai & b+ci \\ -b+ci & di \\\end{bmatrix}

Properties of a Skew Hermitian Matrix

  • The diagonal elements of a skew-Hermitian matrix is either purely imaginary or zero.

For skew-Hermitian matrix,

\(\begin{array}{l}\overline{a_{ij}} = -a_{ij} \text { (1≤ i, j ≤ n)}\end{array} \)

Let aij = a + bi

For diagonal elements, i = j

\(\begin{array}{l}\text { Then, }\overline{a_{ii}} = -a_{ii}\end{array} \)
(1≤ i ≤ n)

\(\begin{array}{l}\overline{a_{ii}} + a_{ii} = 0\end{array} \)

⇒ Real part of aii = 0

  • For k be any real saclar, if A is a skew-Hermitian matrix then (kA)* = −kA.
  • For any real scalars a and b, if matrices A and B are skew-Hermitian then aA +bB is also skew Hermitian.
  • Matrix A is skew-hermitian if and only if iA is Hermitian.

Let A is skew-Hermitian, then we prove that iA is Hermitian

\(\begin{array}{l}\text{Now,  }(iA)^{*} = \bar{i}A^{*} = (-i)(-A) = iA\end{array} \)
  • If A and B are Hermitian Matrix, then AB − BA is skew-Hermitian.

(AB − BA)* = (AB)* − (BA)* = B*A* − A*B* = BA − AB {since A and B are Hermitian}

= −(AB − BA)

  • Ever skew-Hermitian matrix is Normal.

Let A be a skew-Hermitian matrix, to prove A be a Normal matrix we must show

A*A − AA* = 0

Now A*A − AA* = −AA − (−AA) = −A2 + A2 = 0

Note: But every normal matrix is not skew-Hermitian.

  • Conjugate of a skew-Hermitian matrix is also skew-Hermitian.
  • Transpose of a skew-Hermitian matrix is also skew-Hermitian.
  • The trace of a skew-Hermitian matrix is either imaginary or zero.
  • If A is any square matrix, then A − A* is a skew-Hermitian Matrix.
  • Determinant of a skew-Hermitian matrix of odd order is zero.

Let A be a skew-Hermitian matrix of order n such that n is odd.

Now we know, det(A) = det(A*) and det(kA) = kndet(A) where k is any scalar and n is order of A. Using these two properties we have,

det(A) = det(A*) = det(−A) = det[(−1)A] = −1n det(A)

⇒ det(A) = – det(A) {since n is odd}

⇒ 2 det(A) = 0 ⇒ det(A) = 0.

  • Every square matrix A can be represented as the sum of a Hermitian and a skew-Hermitian matrix.

Let A be a square matrix, then A = ½(A + A*) + ½(A − A*) where A + A* is Hermitian and

A − A* is skew-Hermitian.

  • Eigenvalues of skew-Hermitian matrix is either purely imaginary or zero.

Let A be a skew-Hermitian matrix, then A* = −A and let λ be the eigenvalue of A and X be the corresponding eigen vector.

So AX = λX {by definition of eigenvalue and eigen vector)

Multiply X* on both sides we get

X*AX = X*λX

⇒ ((X*A)*)*X = X*λX {since (B*)* = B}

⇒ (A*(X*)*)*X = X*λX { Reversal law of transpose conjugate}

⇒ (−AX)*X = X*λX

⇒ (−λX)*X = X*λX {since AX = λX}

⇒ −X*λ*X = X*λX

\(\begin{array}{l}\Rightarrow -\bar{\lambda} X^{*}X = \lambda X^{*}X\end{array} \)
\(\begin{array}{l}\Rightarrow(\bar{\lambda} + \lambda) X^{*}X = 0\end{array} \)

Now X*X = ||X|| is always positive. Therefore

\(\begin{array}{l}(\bar{\lambda} + \lambda) = 0\end{array} \)
\(\begin{array}{l}\Rightarrow (\bar{\lambda} = – \lambda) \end{array} \)

Thus, λ is either zero or pure imaginary.

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Solved Examples on Skew Hermitian Matrix

Example 1:

\(\begin{array}{l}\text{If } A=\begin{bmatrix}i & 1+i & 2-3i \\-1+i & 2i & 1 \\-2-3i & -1 & 0 \\\end{bmatrix}, \text{ then prove that conjugate of A is skew-Hermitian.}\end{array} \)

Solution:

Clearly, A is a skew-Hermitian matrix and by the properties of skew-Hermitian matrix, conjugate of a A is also skew-Hermitian.

Example 2:

\(\begin{array}{l}\text{Check whether matrix  }\begin{bmatrix}3 & 1+i & 2-3i \\-1+i & 2i & 1 \\-2-3i & -1 & 0 \\\end{bmatrix} \text{ is a skew-Hermitian matrix.}\end{array} \)

Solution:

We know a skew-Hermitian matrix have diagonal elements either zero or pure imaginary. But in the given matrix first diagonal element is neither 0 nor imaginary.

Hence it is not skew-Hermitian.

Frequently Asked Questions on Skew Hermitian Matrix

Q1

How is skew-Hermitian matrix different from Hermitian matrix?

A Hermitian matrix is equal to its conjugate transpose whereas a skew-Hermitian matrix is equal to negative of its conjugate transpose.

Q2

How do you create a skew-Hermitian matrix?

All the diagonal elements should be either zero or pure imaginary. The elements of the ith row and jth column should have real part with opposite signs and same imaginary parts.

Q3

What is the determinant of a skew-Hermitian matrix?

The determinant of skew-Hermitian matrix of odd order is zero.

Q4

What is trace of a skew-Hermitian matrix?

Trace of a skew-Hermitian matrix is either purely imaginary or zero.

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