Let $U$ be an $N\times N$ matrix whose entries are all equal to 1, and let B an $N\times N$ symmetric matrix whose entries are all positive. B is in general non-invertible.
I am interested in the following quantity:
$$A(x) = \|(U + xB)^{-1}\|$$
where $x$ is a positive scalar and norm can be any norm (Frobhenius, largest eigenvalue, ...). I would pick the one that is easiest to work with.
Because $U$ has all entries equal to 1, $U + xB$ is not invertible, my goal is to derive the asymptotic behavior of $A(x)$ for $x\rightarrow0$.
My guess is that $A(x)$ will go to infinity like some power of $\frac{1}{x}$. How could I proceed?
I thank you in advance. My best regards.
First, note that $$ \left(U+xB\right)^{-1}=\frac{1}{x}\left(\frac{1}{x}U+B\right)^{-1}. $$ Since $U=ee^{\intercal}$ where $e$ is the vector of ones, assuming $B$ is invertible, the Sherman-Morrison formula yields $$ \left(\frac{1}{x}U+B\right)^{-1}=B^{-1}-\frac{\frac{1}{x}B^{-1}UB^{-1}}{1+\frac{1}{x}e^{\intercal}B^{-1}e}=B^{-1}-\frac{B^{-1}UB^{-1}}{x+e^{\intercal}B^{-1}e} $$ If $e^\intercal B^{-1} e \neq 0$, taking norms of both sides and applying the triangle inequality yields that the above quantity is bounded as $x\rightarrow0$. In this case, $$ A(x) =\Theta(x^{-1}) \qquad \text{as }x\rightarrow0. $$ If $e^\intercal B^{-1} e = 0$, a similar argument yields $$ A(x) =\Theta(x^{-2}) \qquad \text{as }x\rightarrow0. $$ The different rates of convergence in the two cases are made clear in this plot: