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Is $ \rho (AA^*A)=\rho(A)$ always true?

Here $\rho $ indicates rank of the matrix, and $A$ has entries from $\mathbb{C}$

I found that by multiplication inequality, $ \rho (AA^*A)\leq\rho(A)$. Now I am trying to disprove the strict inequality. $\rho(AA^*)=\rho(A)$ is all I know, but I can't proceed further. Please help me prove or disprove

user511110
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1 Answers1

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Hint: $A^*A$ is Hermitian, so you have a basis of eigenvectors, and the kernel of $A^*A$ is the same as the kernel of $A$

Exodd
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