Question

If $A$ is a skew Hermitian matrix, then the main diagonal elements of $A$ are all

• A. purely real
• B. positive
• C. negative
• D. purely imaginary

Answer 1

Answer: (D)

$A^{\ast }=-A=\Rightarrow (\bar{A}{)}^{\prime }=-A\Rightarrow \overline{a_{ii}}=-a_{ii}\forall i$
$\Rightarrow 2\mathrm{Re}\left(a_{ii}\right)=0\Rightarrow a_{ii}\text{ is purely imaginary. }$