Question

Diagonal elements of a skew-Hermitian matrix are:

• A. Complex numbers of the form $a + ib, a \neq 0, b \neq 0$.
• B. Purely real numbers or zero.
• C. Purely imaginary or zero.
• D. Zero only.

Hint:

Matrix Property Summary: - Hermitian matrix: Diagonal elements are purely real. - Skew-Hermitian matrix: Diagonal elements are purely imaginary or zero. - Skew-symmetric matrix: Diagonal elements are strictly zero.

Answer 1

The Correct Option is C

Step 1: Concept
A square matrix $A \in \mathbb{C}^{n \times n}$ is called a skew-Hermitian matrix if it is equal to the negative of its conjugate transpose (also known as the Hermitian adjoint): \[ A^{\dagger} = -A \quad \text{or} \quad (\bar{A})^T = -A \] where $\bar{A}$ denotes the complex conjugate and $T$ denotes the transpose.

Step 2: Key Formulas and Approach
Let $A = (a_{ij})_{n \times n}$. By definition of skew-Hermitian matrices: \[ a_{ij} = -\bar{a}_{ji} \quad \text{for all } i, j \] For the main diagonal elements, set $i = j$: \[ a_{ii} = -\bar{a}_{ii} \]

Step 3: Step-by-step Explanation

• Let the diagonal entry be represented as $a_{ii} = x + iy$, where $x, y \in \mathbb{R}$.

• The complex conjugate is $\bar{a}_{ii} = x - iy$.

• Substituting this into the skew-Hermitian property $a_{ii} = -\bar{a}_{ii}$: \[ x + iy = -(x - iy) \] \[ x + iy = -x + iy \]
• Subtracting $iy$ from both sides: \[ x = -x \implies 2x = 0 \implies x = 0 \]
• Since $x = 0$, the diagonal element $a_{ii}$ simplifies to: \[ a_{ii} = 0 + iy = iy \]
• If $y = 0$, then $a_{ii} = 0$ (zero).

• If $y \neq 0$, then $a_{ii} = iy$ is purely imaginary.

• Therefore, the diagonal elements of any skew-Hermitian matrix must be either purely imaginary or zero.

Step 4: Final Answer
The diagonal elements of a skew-Hermitian matrix are purely imaginary numbers or zero. Thus, Option (C) is correct.