An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra
Abstract
:1. Introduction
2. MaxPlus Algebra
Algorithm 1 Eigenproblems in MaxPlus Algebra 

3. Latin Square
Algorithm 2 Eigenproblems for Latin Square 

Algorithm 3 Eigenproblems for MaxPlus Algebra 

 1.
 By Algorithm 2, the maximal number in M is the eigenvalue, $i.e.$,$$\lambda =max\left(M\right)=3.$$Consequently,$${M}_{\lambda}=\left[\begin{array}{cccc}1& 2& 0& \u03f5\\ 2& \u03f5& 1& 0\\ 0& 1& \u03f5& 2\\ \u03f5& 0& 2& 1\end{array}\right].$$Next, consider the initial vector$${u}^{*}\left(0\right)=\left(\right)open="("\; close=")">\begin{array}{c}0\\ \u03f5\\ \u03f5\\ \u03f5\end{array}$$After iterating $\left(3\right)$, the following sequence is obtained:$$\begin{array}{ccc}\hfill {u}^{*}\left(1\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(3\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}& \to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(4\right)& ={u}^{*}\left(2\right)\hfill \\ \hfill \left(\right)open="("\; close=")">\begin{array}{c}1\\ 2\\ 0\\ \u03f5\end{array}\to \left(\right)open="("\; close=")">\begin{array}{c}0\\ 1\\ 1\\ 2\end{array}& \to \left(\right)open="("\; close=")">\begin{array}{c}1\\ 2\\ 0\\ 1\end{array}\end{array}\to \left(\right)open="("\; close=")">\begin{array}{c}0\\ 1\\ 1\\ 2\end{array}$$It follows that $s=2,r=4$. To compute a corresponding eigenvector we proceed with the computation of the vector v.$$\begin{array}{ccc}\hfill v& =& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(s\right)\oplus \dots \oplus {u}^{*}(r1)\hfill \\ & =& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\oplus \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(3\right)\hfill \\ & =& \left(\right)open="("\; close=")">\begin{array}{c}0\\ 1\\ 1\\ 2\end{array}\oplus \left(\right)open="("\; close=")">\begin{array}{c}1\\ 2\\ 0\\ 1\end{array}\hfill & =\left(\right)open="("\; close=")">\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\\ .\end{array}$$It is easy to check whether or not $M\otimes v=\lambda \otimes v$. It follows that$$M\otimes v=\left(\right)open="("\; close=")">\begin{array}{c}3\\ 2\\ 3\\ 2\end{array}$$Hence for the eigenvalue $\lambda =3$, the corresponding eigenvector v of the matrix M resulting from the above algorithm is a correct eigenvector.
 2.
 For Algorithm 3, consider the initial vector$$u\left(0\right)=\left(\right)open="("\; close=")">\begin{array}{c}0\\ \u03f5\\ \u03f5\\ \u03f5\end{array}$$Iterating $\left(1\right)$, we obtain$$\begin{array}{ccc}\hfill u\left(1\right)\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\left(3\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}& \to u\left(4\right)& =u\left(2\right)\hfill \\ \hfill \left(\right)open="("\; close=")">\begin{array}{c}2\\ 1\\ 3\\ \u03f5\end{array}\to \left(\right)open="("\; close=")">\begin{array}{c}6\\ 5\\ 5\\ 4\end{array}& \to \left(\right)open="("\; close=")">\begin{array}{c}8\\ 7\\ 9\\ 8\end{array}\end{array}\to \left(\right)open="("\; close=")">\begin{array}{c}12\\ 11\\ 11\\ 12\end{array}$$It follows that $q=2,p=4$. The vector v resulting from Algorithm 3, is given as$$\begin{array}{ccc}\hfill v& =& {\oplus}_{j=1}^{pq}({\lambda}^{\otimes (pqj)}\otimes u(q+j1))\hfill \\ & =& {\oplus}_{j=1}^{2}({\lambda}^{\otimes (2j)}\otimes u(2+j1))\hfill \\ & =& \lambda \otimes u\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\oplus \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\left(3\right)\hfill \\ & =& 3\otimes \left(\right)open="("\; close=")">\begin{array}{c}6\\ 5\\ 5\\ 4\end{array}\oplus \left(\right)open="("\; close=")">\begin{array}{c}8\\ 7\\ 9\\ 8\end{array}\hfill \end{array}& =& \left(\right)open="("\; close=")">\begin{array}{c}9\\ 8\\ 8\\ 7\end{array}\oplus \left(\right)open="("\; close=")">\begin{array}{c}8\\ 7\\ 9\\ 8\end{array}\hfill & =\left(\right)open="("\; close=")">\begin{array}{c}9\\ 8\\ 9\\ 8\end{array}\\ .$$
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Umer, M.; Hayat, U.; Abbas, F. An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra. Symmetry 2019, 11, 738. https://doi.org/10.3390/sym11060738
Umer M, Hayat U, Abbas F. An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra. Symmetry. 2019; 11(6):738. https://doi.org/10.3390/sym11060738
Chicago/Turabian StyleUmer, Mubasher, Umar Hayat, and Fazal Abbas. 2019. "An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra" Symmetry 11, no. 6: 738. https://doi.org/10.3390/sym11060738