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<sup>b</sup> <sup>¼</sup> 2, and <sup>c</sup> <sup>¼</sup> 3, it is easy to show that the eigenvalues are <sup>λ</sup><sup>1</sup> <sup>¼</sup> 0 and <sup>λ</sup>2,<sup>3</sup> ¼ �<sup>3</sup> � <sup>i</sup> ffiffiffi

Is it possible to find an MP-matrix with an eigenvalue λ<sup>i</sup> ¼ 0 such that AMð Þ0 > 1?

Is it possible to find an MP-matrix with complex eigenvalues in an open system?

Other questions are not so simple like the previous one. The next two examples propose

Finally, it is interesting to observe that all cases analyzed here with λ<sup>i</sup> ¼ 0 correspond to closed systems. Moreover, in a previous book chapter [6], it was proved that Reð Þ λ<sup>i</sup> ≤ 0 , ∀i, in any MP open system with three tanks or less. Taking into account all these facts, it can be conjectured that in an open system, all the MP-matrix eigenvalues have negative real part and as a

Mixing problems are interesting sources for applied research in mathematical modeling, ODE, and linear algebra, and—as it was shown—their behavior depends on how they are connected. It has been proved that null eigenvalues are not expected in open systems with three or less components, and is a general conclusion for open MP-matrices that can be

As a final remark, all the MP differential equation systems considered in this chapter have stable or asymptotically stable solutions. Nevertheless, this situation may change depending

prove that null and/or complex eigenvalues are possible.

Is it possible to find an MP-matrix such that AMð Þ0 > GMð Þ0 ?

consequence, all the solutions are asymptotically stable.

obtained by applying the Gershgorin circle theorem.

challenging problems that deserve to be studied:

Figure 9. Three tanks with all the possible connections.

56 Matrix Theory-Applications and Theorems

Question 1:

Question 2:

Question 3:

8. Conclusions

2 <sup>p</sup> , which Victor Martinez-Luaces

Address all correspondence to: victoreml@gmail.com

Faculty of Engineering, UdelaR, Montevideo, Uruguay
