Why do antisymmetric eigenvalues have to be purely imaginary?

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Antisymmetric eigenvalues must be purely imaginary due to the properties of eigenvalue equations involving antisymmetric matrices. The proof presented utilizes the relationship Ax = ax, leading to the conclusion that a = -b, where b is the conjugate of a. This establishes that if A is antisymmetric (A^t = -A), then the eigenvalues must take the form a = x + iy, ensuring that the real part is zero. Thus, antisymmetric matrices inherently yield purely imaginary eigenvalues.

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Why do they have to purely imaginary?

I got a proof that looks like Ax=ax
where a = eigenvalue

therefore Ax.x = ax.x = a|x|^2

Ax.x = x.(A^t)x
where A^t = transpose = -A
x.(-A)x = -b|x|^2

therefore a=-b, where b = conjugate of a

Now is this as far as i need to go?
 
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