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
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 $A$