Question

A $4 \times 4$ matrix $M$ has the property $M^{\dagger} = -M$ and $M^4 = 1$, where $1$ is the $4 \times 4$ identity matrix. Which one of the following is the CORRECT set of eigenvalues of the matrix $M$?

• A. (1, 1, -1, -1)
• B. (i, i, -i, -i)
• C. (i, i, i, -i)
• D. (1, 1, -i, -i)

Hint:

For skew-Hermitian matrices with $M^4 = 1$, the eigenvalues must be purely imaginary and also satisfy $\lambda^4 = 1$, leading to eigenvalues like $i$ and $-i$.

Answer 1

The Correct Option is B

Given the properties $M^{\dagger} = -M$ and $M^4 = 1$, we infer that $M$ is a skew-Hermitian matrix. The eigenvalues of a skew-Hermitian matrix must be purely imaginary. Additionally, the condition $M^4 = 1$ implies that the eigenvalues of $M$ must satisfy $\lambda^4 = 1$, which means the eigenvalues are the fourth roots of unity. The fourth roots of unity are $1, -1, i, -i$.
The eigenvalues of $M$ must be purely imaginary and satisfy $\lambda^4 = 1$. Therefore, the eigenvalues of $M$ are $(i, i, -i, -i)$. Hence, the correct answer is (B).