In my ODEs class, we have the defined the Lozinskii Measure by
$\displaystyle\mu(A) = \lim_{h\to 0^{+}}\frac{\|I + hA\| - 1}{h}$, where $\|\cdot\|$ is the matrix norm defined by
$\displaystyle\|A\| = \sup_{|x|\leq 1} |Ax|$ and $|\cdot|$ is any norm on $\mathbb{R}^{n}$.
I am asked to show several properties of this, the first of which is:
a) $\mu(A)$ exists for any $n\times n$ matrix $A$.
All I have worked out is this:
For fixed $h > 0$ (to avoid writing limit) I define $\mu_{h}(A) = \frac{|I + hA| - 1}{h}$. Then if I apply the definition of matrix norm:
\begin{align*} \mu_{h}(A) &= \frac{\sup_{|x|\leq 1}|(I + hA)x| - 1}h\\ &= \frac{\sup_{|x|\leq 1}|x + hAx| - 1}h.\end{align*}
But this gets me nowhere without any insight on the norm $|\cdot|$.
I'm stuck here and I was hoping someone might have a pointer for me?