Skew-Hermitian Operators

Question

Skew-Hermitian operators:
Determine the properties of the eigenvalues and eigenvectors of a skew-Hermitian operator. Are the eigenvalues real, complex, purely imaginary? Are the eigenvectors orthogonal?
Heisenberg uncertainty principle:
Derive the Heisenberg uncertainty principle for the following cases:
a) A and B operators are Hermitian but the quantum state is mixed (not pure)
b) A is Hermitian and and B is skew-Hermitian, the quantum state is pure.
What if A and B are both skew-Hermitian?

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