Question

If the Eigenvalues of Skew-Hermitian matrices and Eigenvalues of Hermitian matrices are plotted on Argand plane, then the number of points having amplitude $\frac{7\pi}{4}$ is

• A. 1
• B. 0
• C. more than 4
• D. infinite in number

Hint:

Use the geometric interpretation of eigenvalues of Hermitian and Skew-Hermitian matrices to understand their location on Argand plane.

Answer 1

The Correct Option is B

- Hermitian matrices have real eigenvalues, which lie on the real axis in the Argand plane ⇒ amplitude = 0 or \(\pi\).
- Skew-Hermitian matrices have purely imaginary eigenvalues, which lie on the imaginary axis ⇒ amplitude = \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\).
Hence, no eigenvalue lies at amplitude \(\frac{7\pi}{4}\).