I am new to this website application and I don't know if I can be able to interact with you and ask for help. I hope somebody here can help me with this question.
Explain why the $\ell_1$-norm on $M_n$ is a matrix norm that is not an induced norm.
I don't understand the problem. I thought that the $\ell_1$ norm on $M_n$ is already an induced norm? Can somebody help me with this? I will truly appreciate it. Can somebody show this to me, because I already believe a lie that the $\ell_1$ on an $n \times n$ matrix is a matrix norm and so is an induced norm.
The $\ell_1$-norm is defined as
$$\displaystyle{\|A\|_{\ell_1} := \sum_{i,j=1}^n |a_{ij}|}$$
Best Answer
Any induced norm $\|\;\|^*$ on $M_n(\mathbb{C})$ (or $M_n(\mathbb{R})$) satisfies the submultiplicative property: $$\|AB\|^*\leq \|A\|^*\|B\|^*$$ and $$\|I_n\|^*=1 $$ where $I_n$ is the identity matrix in $M_n(\mathbb{C})$.
Let $$\|A\|_{\ell_1}:=\sum_{1\leq I,j\leq n}|a_{ij}|$$
This is certainly a norm in $M_n(\mathbb{C})$. But $\|I_n\|_{\ell_1}=n>1$ (for $n>1$)
Edit: Interestingly enough, $\|\;\|_{\ell_1}$ does satisfy the sub-multiplicative property.
There is also the slew of norms $$\|A\|_{\ell_p}=\Big(\sum_{ij}|a_{ij}|^p\Big)^{1/p},\qquad p\geq1$$ Since $\|I_n\|_{\ell_p}=n^{1/p}>1$ ($n>1$), none of these norms are induced (or subordinated) to any norm in $\mathbb{C}^n$. For $1\leq p\leq 2$, the norms $\|\;\|_{\ell_p}$ are sub-multiplicative, for $p>2$ the are not.