Let
be given by:
![{\displaystyle {\begin{pmatrix}0&{\frac {1}{2}}&{\frac {1}{4}}\\{\frac {5}{7}}&0&{\frac {1}{7}}\\{\frac {3}{10}}&{\frac {3}{5}}&0\end{pmatrix}}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/298f9b73421349a55c488f285ae3bc674ac97d58)
We need to show that C is smaller than unity in some norm. Therefore, we calculate:
![{\displaystyle {\begin{aligned}||C||_{\infty }&=\max _{i}\sum _{j}|c_{ij}|=\max \left\lbrace {\frac {3}{4}},{\frac {6}{7}},{\frac {9}{10}}\right\rbrace ={\frac {9}{10}}<1.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7329323f519d183b94974842ec08b9685085a6f8)
Thus, we know from the statement above that
exists.
The Neumann series has been used for linear data detection in massive multiuser multiple-input multiple-output (MIMO) wireless systems. Using a truncated Neumann series avoids computation of an explicit matrix inverse, which reduces the complexity of linear data detection from cubic to square.[1]
Another application is the theory of Propagation graphs which takes advantage of Neumann series to derive closed form expression for the transfer function.
- Werner, Dirk (2005). Funktionalanalysis (in German). Springer Verlag. ISBN 3-540-43586-7.