Skew Hermitian Matrix 

A Skew Hermitian Matrix is a square matrix A that satisfies the condition A* = –A, where A* represents the conjugate transpose of A. In other words, each element of the matrix satisfies $a_{ij}=-\overline{a_{ji}}$ , where $\overline{a_{ji}}$ denotes the complex conjugate of the element a_ji. Skew-Hermitian matrices have purely imaginary diagonal elements and are used in various applications in quantum mechanics and linear algebra.

1.0Skew Hermitian Matrix Definition

A matrix A is called a Skew Hermitian Matrix if it satisfies the condition:

A^*=-A

Where A* denotes the conjugate transpose (also known as the Hermitian transpose) of the matrix A, and −A is the matrix with all elements negated. This means that for any Skew Hermitian matrix, its conjugate transpose is the negative of the matrix itself. In simpler terms, a Skew Hermitian matrix is a matrix whose entries are the negatives of the conjugates of the corresponding entries when transposed.

2.0Skew Hermitian Matrix Formula

The general formula for a Skew Hermitian matrix can be written as:

$A=\left(\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$ 

For the matrix to be Skew Hermitian, the elements must satisfy the following:

1. Diagonal Elements: The diagonal elements of a Skew Hermitian matrix must be purely imaginary numbers (or zero).

$a_{ii}=-\overline{a_{ii}}$

This condition ensures that the diagonal elements are either 0 or a purely imaginary number, where $\overline{a_{ii}}$ is the complex conjugate of a_ii.

2. Off-diagonal Elements: For the off-diagonal elements a_ij and a_ji, the following condition holds:

$a_{ij}=-\overline{a_{ji}}$

This means the elements at position (i, j) are the negatives of the conjugates of the corresponding elements at position (j, i).

3.0Skew Hermitian Matrix Properties

Skew Hermitian matrices have several interesting properties:

1. Purely Imaginary Diagonal Entries: The diagonal elements of a Skew Hermitian matrix are either zero or purely imaginary.
2. Conjugate Transpose: As per the definition, A^*=-A, meaning the conjugate transpose of a Skew Hermitian matrix is the negative of the matrix. 
3. Eigenvalues: The eigenvalues of a Skew Hermitian matrix are always purely imaginary. This is because the matrix satisfies the condition A^*=-A, which implies that the eigenvalues must lie along the imaginary axis in the complex plane.
4. Trace: The trace of a Skew Hermitian matrix is always zero because the diagonal entries are either zero or purely imaginary numbers, and their sum must cancel out.
5. Real Symmetric Matrices: Skew Hermitian matrices are closely related to Hermitian matrices, but unlike Hermitian matrices, which have real eigenvalues, Skew Hermitian matrices have purely imaginary eigenvalues. 

4.0Skew Hermitian Matrix Condition

The condition for a matrix to be Skew Hermitian is:

A* = –A 

For example, let’s check whether a given matrix satisfies the Skew Hermitian condition.

Consider the matrix:

$A=\left(\begin{array}{ccc}0& i& 2i\\ -i& 0& 3i\\ -2i& -3i& 0\end{array}\right)$

To check if A is Skew Hermitian, we compute its conjugate transpose A* and verify if it equals −A.

$A^{\ast }=\left(\begin{array}{ccc}0& i& -2i\\ -i& 0& -3i\\ 2i& 3i& 0\end{array}\right)$

Since A* = –A, the matrix satisfies the Skew Hermitian condition.

5.0Skew Hermitian Matrix Eigenvalue

As mentioned earlier, the eigenvalues of a Skew Hermitian matrix are purely imaginary. To find the eigenvalues of a Skew Hermitian matrix, we typically solve the characteristic equation:

$\det (A-\lambda I)=0$

Where λ is an eigenvalue and I is the identity matrix. For a Skew Hermitian matrix, the solution to this characteristic equation will always yield purely imaginary eigenvalues. 

6.0Hermitian and Skew-Hermitian Matrix

While Skew Hermitian matrices are closely related to Hermitian matrices, they have distinct differences:

• A Hermitian Matrix satisfies the condition A* = A, meaning its conjugate transpose is equal to the matrix itself. Hermitian matrices have real eigenvalues and are often used to represent self-adjoint operators in quantum mechanics.
• A Skew Hermitian Matrix, on the other hand, satisfies the condition A* = –A, and its eigenvalues are purely imaginary.

7.0Solved Example of Skew Hermitian Matrix

Example 1: Consider the matrix A:

$A=\left(\begin{array}{cc}0& 2-3i\\ -2+3i& 0\end{array}\right)$

Check the matrix is Skew Hermitian and Find the eigenvalue.

Solution: 

Step 1: Check if the matrix is Skew Hermitian

To check if the matrix is Skew Hermitian, we need to verify that A* = –A, where A* is the conjugate transpose of A.

1. First, find the conjugate transpose A*:

$A^{\ast }=\left(\begin{array}{cc}0& -2+3i\\ 2-3i& 0\end{array}\right)$

2. Now, check if A* = –A:

$-A=\left(\begin{array}{cc}0& -2+3i\\ 2-3i& 0\end{array}\right)$ 

Since A* = –A, this matrix is indeed a Skew Hermitian Matrix.

Step 2: Find the eigenvalues of A

To find the eigenvalues of the matrix, we solve the characteristic equation $\det (A-\lambda I)=0$, where I is the identity matrix and λ represents the eigenvalues.

$\begin{array}{rl}& A-\lambda I=\left(\begin{array}{cc}-\lambda & 2-3i\\ -2+3i& -\lambda \end{array}\right)\\ & \text{ The determinant of }A-\lambda I\text{ is: }\\ & \begin{array}{rl}& \det (A-\lambda I)=(-\lambda )(-\lambda )-(2-3i)(-2+3i)\\ & ={\lambda }^2-[(2)(-2)+(-3i)(3i)]\\ & ={\lambda }^2-[-4+9]\\ & ={\lambda }^2+5\end{array}\end{array}$

Set the determinant equal to zero:

$\begin{array}{rl}& {\lambda }^2+5=0\\ & {\lambda }^2=5\lambda \\ & =\pm i\sqrt{5}\end{array}$

Thus, the eigenvalues are $\lambda =\pm i\sqrt{5}$, which are purely imaginary, as expected for a Skew Hermitian matrix.

Example 2: Verify if the following matrix is skew-Hermitian:

$A=\left(\begin{array}{ccc}0& 2-i& 4+3i\\ -(2+i)& 0& 5\\ -(4-3i)& -5& 0\end{array}\right)$

Solution:

To verify if matrix A is skew-Hermitian, we need to check if A* = –A, where A* is the conjugate transpose of A.

Step 1: Find the conjugate transpose of A (i.e., A*).

1. Take the transpose of A.
2. Replace each element with its complex conjugate.

The original matrix A is:

$A=\left(\begin{array}{ccc}0& 2-i& 4+3i\\ -(2+i)& 0& 5\\ -(4-3i)& -5& 0\end{array}\right)$ 

• The transpose of A (denoted as $A^T$) is:

$A^T=\left(\begin{array}{ccc}0& -(2+i)& -(4-3i)\\ 2-i& 0& -5\\ 4+3i& 5& 0\end{array}\right)$ 

• Now, take the complex conjugate of each element of $A^T$:

$A^{\ast }=\left(\begin{array}{ccc}0& -(2-i)& -(4+3i)\\ 2+i& 0& -5\\ 4-3i& -5& 0\end{array}\right)$ 

Step 2: Check if A* = –A.

Now, compare A* and −A:

$\begin{array}{rl}& A^{\ast }=\left(\begin{array}{ccc}0& -(2-i)& -(4+3i)\\ 2+i& 0& -5\\ 4-3i& -5& 0\end{array}\right)\\ & -A=\left(\begin{array}{ccc}0& -(2-i)& -(4+3i)\\ 2+i& 0& -5\\ 4-3i& -5& 0\end{array}\right)\end{array}$

We see that A* = –A, thus confirming that A is a skew-Hermitian matrix.

Example 3: Verify if the following matrix is skew-Hermitian:

$A=\left(\begin{array}{ccc}0& 1+i& 2\\ -1-i& 0& 3\\ -2& -3& 0\end{array}\right)$

Solution:

1. Find A* (conjugate transpose of A):
2. • Take the transpose of A:

$A^T=\left(\begin{array}{ccc}0& -1-i& -2\\ 1+i& 0& -3\\ 2& 3& 0\end{array}\right)$

• Now, take the complex conjugate of each element:

$A^{\ast }=\left(\begin{array}{ccc}0& -1+i& -2\\ 1+i& 0& -3\\ 2& -3& 0\end{array}\right)$

2. Check if A* = –A:

$-A=\left(\begin{array}{ccc}0& -1-i& -2\\ 1+i& 0& -3\\ 2& 3& 0\end{array}\right)$

Clearly, A* = –A, so the matrix is skew-Hermitian.

Example 4: Check if the matrix is skew-Hermitian:

$B=\left(\begin{array}{ccc}0& 3-i& -2+4i\\ -(3+i)& 0& 1\\ 2-4i& -1& 0\end{array}\right)$

Solution:

1. Find B*:
2. • First, transpose B:

$B^T=\left(\begin{array}{ccc}0& -(3+i)& 2-4i\\ 3-i& 0& -1\\ -2+4i& 1& 0\end{array}\right)$

• Take the complex conjugate:

$B^{\ast }=\left(\begin{array}{ccc}0& -(3-i)& 2+4i\\ 3+i& 0& -1\\ -2-4i& 1& 0\end{array}\right)$

2. Check if B* = -B:

$-B=\left(\begin{array}{ccc}0& -(3-i)& 2+4i\\ 3+i& 0& -1\\ -2-4i& 1& 0\end{array}\right)$

Since B* = –B, the matrix is skew-Hermitian.

Example 5: Given the matrix

$C=\left(\begin{array}{ccc}0& 2& -1+2i\\ -2& 0& 1\\ 1-2i& -1& 0\end{array}\right)$

Determine if C is skew-Hermitian.

Solution:

1. Find C*:
2. • First, transpose C:

$C^T=\left(\begin{array}{ccc}0& -2& 1-2i\\ 2& 0& -1\\ -1+2i& 1& 0\end{array}\right)$

• Now, take the complex conjugate:

$C^{\ast }=\left(\begin{array}{ccc}0& -2& 1+2i\\ 2& 0& -1\\ 1-2i& 1& 0\end{array}\right)$

2. Check if C* = –C:

$-C=\left(\begin{array}{ccc}0& -2& -1-2i\\ 2& 0& -1\\ 1+2i& 1& 0\end{array}\right)$

Since $C^{\ast }\ne -C$, the matrix is not skew-Hermitian.

Example 6: Check if the matrix is skew-Hermitian:

$D=\left(\begin{array}{ccc}0& 2i& 3\\ -2i& 0& 1\\ -3& -1& 0\end{array}\right)$

Solution:

1. Find D*:
2. • Transpose D:

$D^T=\left(\begin{array}{ccc}0& -2i& -3\\ 2i& 0& -1\\ 3& 1& 0\end{array}\right)$

• Take the complex conjugate:

$D^{\ast }=\left(\begin{array}{ccc}0& 2i& -3\\ -2i& 0& -1\\ 3& 1& 0\end{array}\right)$

2. Check if D* = –D:

$-D=\left(\begin{array}{ccc}0& -2i& -3\\ 2i& 0& -1\\ 3& 1& 0\end{array}\right)$

Since D* = –D, the matrix is skew-Hermitian.

Example 7: Given the matrix

$B=\left(\begin{array}{ccc}0& 2& -1+2i\\ -2& 0& 1\\ 1-2i& -1& 0\end{array}\right)$

Determine if B is skew-Hermitian.

Solution:

1.Find B*:

First, transpose B:

$B^T=\left(\begin{array}{ccc}0& -2& 1-2i\\ 2& 0& -1\\ -1+2i& 1& 0\end{array}\right)$

• Now, take the complex conjugate:

$B^{\ast }=\left(\begin{array}{ccc}0& -2& 1+2i\\ 2& 0& -1\\ 1-2i& 1& 0\end{array}\right)$

2.Check if B* = –B:

$-B=\left(\begin{array}{ccc}0& -2& -1-2i\\ 2& 0& -1\\ 1+2i& 1& 0\end{array}\right)$

Since $B^{\ast }\ne -B$, the matrix is not skew-Hermitian.

8.0Practice Questions on Skew Hermitian Matrix

Question 1: Given the matrix $A=\left(\begin{array}{cc}0& 3+4i\\ -3-4i& 0\end{array}\right)$, verify if it is a Skew Hermitian matrix and find its eigenvalues.

Question 2: Find the eigenvalues of the Skew Hermitian matrix $A=\left(\begin{array}{cc}0& i\\ -i& 0\end{array}\right)$.

Question 3: Check whether the matrix $A=\left(\begin{array}{ccc}0& 2+4i& -5i\\ -2-4i& 0& 3i\\ 5i& -3i& 0\end{array}\right)$ is Skew Hermitian. If yes, find its eigenvalues.

Question 4: For the matrix $A=\left(\begin{array}{cc}-4i& 3+2i\\ -3+2i& 4i\end{array}\right)$, determine if it is Skew Hermitian and find its eigenvalues.