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- TL;DR Summary
- In Ordinary Differential Equations by Andersson and Böiers, I'm reading about so-called weakly singular systems, that is, ##\pmb{x}'(z)=A(z)\pmb{x}(z)## where ##A(z)## is analytic except at the origin where it has a simple pole (this means all its entries are analytic with at most a simple pole). I'm confused about an estimate made in a proof in this section.

Let ##A(z)## be a matrix function with a simple pole at the origin; in other words, we can expand it into a Laurent series of the form ##\frac1{z}A_{-1}+A_0+zA_1+\ldots##, where ##A_i## are constant matrices and ##A_{-1}\neq 0##. Fix ##\theta_0\in[0,2\pi)## and ##c\in(0,1)## (here ##1## could also be any other real, finite number) and let ##0<s<c##. My textbook claims that $$\lVert A(se^{i\theta_0})\rVert\leq m|se^{i\theta_0}|^{-1}=\frac{m}{s},\qquad 0<s<c,$$ for some ##m>0## and that this should follow from the inequality ##\lVert A\rVert\leq \left(\sum_{j,k=0}^n |a_{jk}|^2\right)^{1/2}##. I do not understand this, because consider for instance $$\begin{bmatrix} \frac1{z}&1\\ 2&3 \end{bmatrix}=\frac1{z}\begin{bmatrix}1&0\\ 0&0\end{bmatrix}+\begin{bmatrix}0&1\\ 2&3\end{bmatrix}$$ I don't see how the claimed inequality follows from ##\lVert A\rVert\leq \left(\sum_{j,k=0}^n |a_{jk}|^2\right)^{1/2}## in this case, since it seems like we can't factor out ##\frac1{s}## from the sum.