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.

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:

Example of a Skew Hermitian Matrix

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

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:

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

Properties of a Skew Hermitian Matrix

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

For skew-Hermitian matrix,

Let a_ij = a + bi

For diagonal elements, i = j

⇒ Real part of a_ii = 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

• 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

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

Thus, λ is either zero or pure imaginary.

Related Articles

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

Solved Examples on Skew Hermitian Matrix

Example 1:

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:

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.