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Suppose $A∈M_n(\mathbb{C})$ is a nonsingular matrix and $\bar{A}$ is conjugate of $A$, i.e. entries of $\bar{A}$ is obtained by replace entries of $A$ with their conjugate. Let $\lambda$ be the real negative eigenvalue of $A\bar{A}$. Prove the algebraic multiplicity of $\lambda$ is even.

I've tried it if every entries of $A$ are real, the algebraic multiplicity of $\lambda$ is $0$. But I can't generalize it if some entries of $A$ are not pure real or pure imaginary.

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    This question has been already asked http://math.stackexchange.com/questions/135954/multiplicity-of-eigenvalues easlier. – 2017-01-27
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    Thank you, I coudn't find it before – 2017-01-28

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