Condition on norm for Neumann series on matrix inverse

Question

I am trying to understand what is the flaw in my following argument. For the identity $(I - A)^{-1} = \sum_{k=0}^{\infty} A^k$ to apply we need to show $\|A\| < 1$.

For some $A$ I could show that $\|A\|_F < \lambda$ for some finite $\lambda$.

To apply the identity, can we use the alternate norm $\|A\|_* = \frac{1}{\lambda} \|A\|_F$?

Answer 1

If $\|\cdot\|$ is a submultiplicative matrix norm, then $\|\cdot\|_* = c\|\cdot\|$ is not necessarily submultiplicative if $c < 1$ because $\|AB\|_* = c\|AB\|\le c\|A\|\|B\|$, but you want $\le c^2\|A\|\|B\|$. In the Neumann theorem the matrix norm is supposed to be submultiplicative in order to have $\|A^k\|\le\|A\|^k$.

Answer 2

The identity holds whenever the spectral radius is less than $1$. If $\lambda$ is large enough such that $A' = A/\lambda$ has spectral radius less than $1$, then the series can be applied to $A'$.

Defining a new norm has no bearing on whether or not the formula applies.