If $\lVert A \rVert$ is a matrix norm, then $\lVert A+A*\rVert$ is also a matrix norm?

Question

I was reading about matrix norms and I found this statement. Any ideas how this one can be proved? A* is the complex conjugate transpose of A. In a simpler question, is even $\lVert A*\rVert$ a matrix norm?

Answer 1

Since $AA^*$ and $A^*A$ have the same non-zero eigenvalue $\lambda$, as a result, $\|A^*\|^2=\lambda_{max}(AA^*)=\lambda_{max}(A^*A)=\|A\|^2$ and thus $\|A\|=\|A^*\|$. From sub-additive property of norm $\|A+A^*\|\leq{\|A\|}+\|A^*\|$ implies $\|A+A^*\|\leq{2\|A\|}$.