TITLE: Matrix norm question
QUESTION [5 upvotes]: Let $A^*$ denote the complex conjugate transpose of a matrix $A$. In the Euclidean norm, if
$$||A^*A+AA^*||=||A^*A||$$
does it imply that $AA^*=0$. If not, could you give a counter-example?

REPLY [3 votes]: For a counterexample, consider
$$
A:=\pmatrix{0&1\\0&0}.
$$