In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on ${ \mathbb C}^n$ such that $$N(A)=\max\{ \|Ax\|_2: \|x\|_1=1, x\in { \mathbb C}^n\}$$ for all $A \in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.
Categories: math.FA