Consider the matrix $$D=\begin{pmatrix}1&0\\0&e^{i\theta}\end{pmatrix}.$$ For the commonly used norms $\|\cdot\|$ on $\mathbb{C}^2$ or for $\theta=0$ the associated subordinate norm is $1$. Is it always true ? can a subordinate norm be strictly bigger than one ?



Existence of weird complex norms
Consider the norm $\Vert\cdot\Vert$ on $\mathbb{C}^2$ defined by $\Vert z \Vert^2 := |z\_1|^2+|z\_1+z\_2|^2$. Then
$$\left\Vert \left(\begin{array}{c}1 \\ -1 \end{array}\right) \right\Vert^2 = 1,$$
$$\left\Vert D\left(\begin{array}{c}1 \\ -1 \end{array}\right) \right\Vert^2 = \left\Vert \left(\begin{array}{c}1 \\ -e^{i\theta} \end{array}\right) \right\Vert^2 = 1 + |1-e^{i\theta}|^2.$$
If $e^{i\theta} \ne 1$, we derive that the subordinate norm of $D$ is $>1$.


