Rivalry2

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Original 2026-01-01

Modified 2026-02-28

1 Comparing A* and -A, we see that the condition A* = -A is satisfied. So, A is a skew Hermitian matrix.

2 The elements of a skew Hermitian matrix follow these conditions:

3 <ol><li>All diagonal elements are either purely imaginary or zero. </li>

4 <li>Non-diagonal elements can have both real and imaginary parts.</li>

5 </ol>Based on these conditions, the general<a>formula</a>of a skew Hermitian matrix, For a 2 × 2 skew Hermitian matrix is:

6 \( A = \begin{bmatrix} xi & y + zi \\ -y + zi & wi \end{bmatrix} \)

7 For a 3 × 3 skew Hermitian matrix:

8 \( A = \begin{bmatrix} ai & b + ci & c + di \\ -b + ci & ei & g + hi \\ -c + di & -g + hi & ki \end{bmatrix} \)

9 What are the Properties of a Skew Hermitian Matrix?

10 Key properties of a skew Hermitian matrix include:

11 <ol><li>All diagonal elements are purely imaginary or zero, i.e., of the form bi, where b R. </li>

12 <li>The element at position (i, j) is the negative of the complex conjugate of the element at position (j, i) in a skew Hermitian matrix. aij = - aij -. </li>

13 <li>The eigenvalues of a skew Hermitian matrix are either zero or purely imaginary. </li>

14 <li>These matrices are always square. </li>

15 <li>If all entries in a matrix are real<a>numbers</a>, then a skew Hermitian matrix becomes a skew-<a>symmetric matrix</a>. This is because taking the complex conjugate does not affect real numbers. </li>

16 <li>The<a>sum</a>of two skew Hermitian matrices is also skew Hermitian. </li>

17 <li>If A is skew Hermitian and α∈R then αA is also skew Hermitian. </li>

18 <li>The zero matrix is a skew Hermitian matrix because 0* = -0 = 0. </li>

19 <li>When a skew Hermitian matrix A, is multiplied by imaginary unit \(i = \sqrt-1\), we get the Hermitian matrix iA.</li>

20 </ol>What is the condition for the Skew Hermitian Matrix?

21 For a matrix to be skew Hermitian, it must satisfy the condition A* = -A. Let us take an example to check for the condition. Let,

22 \( A = \begin{bmatrix} 0 & 3 + 2i \\ -3 + 2i & 0 \end{bmatrix} \)

23 To check if A* = -A, Find transpose AT

24 \( A^{T} = \begin{bmatrix} 0 & -3 + 2i \\ 3 + 2i & 0 \end{bmatrix} \)

25 Complex conjugate A*

26 \( A^{*} = \begin{bmatrix} 0 & -3 - 2i \\ 3 - 2i & 0 \end{bmatrix} \)

27 Now, we find -A

28 \( -A = \begin{bmatrix} 0 & -3 - 2i \\ 3 - 2i & 0 \end{bmatrix} \)

29 Skew Hermitian Matrix Eigenvalue

30 As established earlier, we see that the eigenvalues of a skew Hermitian matrix are purely imaginary or zero. So, to find these eigenvalues, we solve the characteristic<a>equation</a>\( \det(A - \lambda I) = 0 \). Here \(\lambda\) is an eigenvalue, and I is the<a>identity matrix</a>.