Question about skew-Hermitian

Definition. A square matrix
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with complex entries is skew-Hermitian, if

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.
Properties.

The trace of a skew-Hermitian matrix is imaginary.
The eigenvalues of a skew-Hermitian matrix are imaginary.
Proof. Property (1) follows directly from property (2) since the trace is the sum of the eigenvalues. But one can also give a simple proof as follows. Let
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, i.e.,






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(1)
Here,
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. Thus





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Hence the eigenvalue
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is imaginary.

If x is an eigenvector of H with eigenvalue L, then
< Hx, x > = L< x,x >
< Hx, x > = < -H*x, x > = - < x, Hx > = - L*< x, x > where L* = conjugate
comparing, we have L = -L*. so L is pure imaginery.