I am struggling with a problem for the past few days and I would really appreciate someone's help. The problem is the following:
If $A$ is a square $n×n$ matrix, prove that the matrix $$B=\begin{bmatrix}A^*&0\\0&A\end{bmatrix}$$ has the same spectral norm (2-norm) as matrix $A$.
I managed to prove that the eigenvalues of all 4 matrices $(A,A^*,B,B^*)$ are the same, but I am not sure that it means the spectral norm is also the same. Sorry for poor post quality, it is my first time here.