Question

Which of the following cannot be an eigen value for an unitary matrix?

• A. \(\frac{1}{2} + \frac{\sqrt{3}}{2}i\)
• B. \(i\)
• C. \(-i\)
• D. \(\frac{1}{2} + \frac{\sqrt{5}}{2}i\)

Hint:

The eigenvalues of unitary matrices lie on the unit circle in the complex plane.
Always compute \(a^2 + b^2\) to rapidly check if the modulus equals 1.

Answer 1

The Correct Option is D

Concept:
• A matrix \(U\) is called an unitary matrix if \(U^{\dagger}U = I\), where \(U^{\dagger}\) is the conjugate transpose of \(U\).
• The eigenvalues of an unitary matrix always have a absolute value (modulus) of 1.
• For any eigenvalue \(\lambda = a + bi\), its modulus is given by \(|\lambda| = \sqrt{a^2 + b^2} = 1\).

Step 1: Check the modulus of option A
\[ \left|\frac{1}{2} + \frac{\sqrt{3}}{2}i\right| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \] \[ = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

Step 2: Check the modulus of option B and C
\[ |i| = \sqrt{0^2 + 1^2} = 1 \] \[ |-i| = \sqrt{0^2 + (-1)^2} = 1 \]

Step 3: Check the modulus of option D
\[ \left|\frac{1}{2} + \frac{\sqrt{5}}{2}i\right| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{5}}{2}\right)^2} \] \[ = \sqrt{\frac{1}{4} + \frac{5}{4}} = \sqrt{\frac{6}{4}} = \sqrt{\frac{3}{2}} \neq 1 \]