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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?

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For a counterexample, consider $$ A:=\pmatrix{0&1\\0&0}. $$

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  • $\begingroup$ Thank you. This is a part of an other problem, which I posted here: link. $\endgroup$ – PeterA Feb 1 '15 at 14:28

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