Question

show that for a hermitian matrix its diagonal elements are all purely real numbers​

Answer 1

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The diagonal elements of a skew-Hermitian matrix are purely imaginary or zero. You can see this as follows:

A matrix ∈ℂ× is skew-Hermitian if

†=−.

This means that the components of satisfy

=−⎯⎯⎯⎯⎯⎯(,∈{1,...,}).

So for the diagonal elements (=) you get

=−⎯⎯⎯⎯⎯⎯.

Fix ∈{1,...,}. Since ∈ℂ, you can write =+ with ,∈ℝ. Then you get

+=−(−)=−+⇒=−⇒=0.

Hence = and so is either purely imaginare (if ≠0) or zero (if =0). Since this is true for all ∈{1,...,}, the assertion follows.