Theorem 1

The Gershgorin disk corresponding to this row is centered at <sup>a</sup><sup>22</sup> ¼ � ð Þ <sup>Φ</sup>21þΦ<sup>23</sup>

Now, if a flux balance is performed in this internal tank, we have this equation: Φ<sup>12</sup> þ Φ<sup>32</sup> ¼ Φ<sup>21</sup> þ Φ23, and then j j a<sup>22</sup> ¼ R2, and the corresponding Gershgorin disk will look like the one

Finally, if the third ODE of Eq. (17) is considered, this equation can be written as

This output tank equation corresponds to the third row of the MP-matrix Eq. (18):

The Gershgorin disk corresponding to this row is centered at the point <sup>a</sup><sup>33</sup> ¼ � ð Þ <sup>Φ</sup>31þΦ32þΦ<sup>0</sup>

corresponding Gershgorin disk will look like as the one schematized in Figure 7.

The flux balance in this case gives Φ<sup>13</sup> þ Φ<sup>23</sup> ¼ Φ<sup>31</sup> þ Φ<sup>32</sup> þ Φ0, and then j j¼ a<sup>33</sup> R3, and the

Taking into account all these results, the Gershgorin circles for the MP of Figure 5 are shown in

<sup>R</sup><sup>2</sup> <sup>¼</sup> <sup>Φ</sup>12þΦ<sup>32</sup> <sup>V</sup><sup>2</sup> .

dC<sup>3</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>13</sup>

> Φ<sup>13</sup> V3

Figure 8.

schematized in Figure 7.

52 Matrix Theory-Applications and Theorems

<sup>V</sup><sup>3</sup> <sup>C</sup><sup>1</sup> <sup>þ</sup> <sup>Φ</sup><sup>23</sup>

Φ<sup>23</sup> V3

with radius <sup>R</sup><sup>3</sup> <sup>¼</sup> <sup>Φ</sup>13þΦ<sup>23</sup>

<sup>V</sup><sup>3</sup> <sup>C</sup><sup>2</sup> � ð Þ <sup>Φ</sup>31þΦ32þΦ<sup>0</sup>

� ð Þ <sup>Φ</sup><sup>31</sup> <sup>þ</sup> <sup>Φ</sup><sup>32</sup> <sup>þ</sup> <sup>Φ</sup><sup>0</sup> V3

<sup>V</sup><sup>3</sup> .

Figure 7. The Gershgorin disk corresponding to an internal tank.

Figure 8. Gershgorin circles for a three-tank system with recirculation.

<sup>V</sup><sup>3</sup> C3.

.

<sup>V</sup><sup>2</sup> < 0 with radius

<sup>V</sup><sup>3</sup> < 0

In an open system, if the ith tank is an input one, then the diagonal entry of the ith row is aii < 0 and aii j j > Ri being the sum of the non-diagonal entry modules of that row.
