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
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
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Use the geometric interpretation of eigenvalues of Hermitian and Skew-Hermitian matrices to understand their location on Argand plane.
Updated On: May 26, 2025
- 1
- 0
- more than 4
- infinite in number
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The Correct Option is B
Solution and Explanation
- 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 \(\frac{7\pi}{4}\).
- 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 \(\frac{7\pi}{4}\).
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