5. The Gershgorin circle theorem

The Gershgorin circle theorem first version was published by S. A. Gershgorin in 1931 [9]. This theorem may be used to bind the spectrum of a complex matrix, and its statement is the following:

Theorem (Gershgorin)

Once again, the ODE system can be written as , where the MP-matrix is:

It is easy to observe that the outgoing flux Φ<sup>0</sup> only appears in the last entry of the MP-matrix and the incoming flux Φ<sup>0</sup> only is involved in the first entry of vector B. These facts—particularly the first one—are relevant when applying the Gershgorin circle theorem, which will be exposed

The Gershgorin circle theorem first version was published by S. A. Gershgorin in 1931 [9]. This theorem may be used to bind the spectrum of a complex matrix, and its statement is the

In the previous ODE system, the independent vector is:

Figure 5. Three tanks with all the possible connections.

50 Matrix Theory-Applications and Theorems

in the next section.

following:

5. The Gershgorin circle theorem

ð18Þ

ð19Þ

If is an matrix, with entries being , and is the sum of the non-diagonal entry modules in the th row, then every eigenvalue of lies within at least one of the closed disks , called Gershgorin disks.

This theorem was widely used in previous book chapters [6, 10, 11] in order to obtain new results about matrices corresponding to chemical problems.

Here, the main purpose is to apply this theorem to MP-matrices as a method to bind their eigenvalues, depending on the characteristics of the MP ODE system, and, even more, the compartment considered.

For instance, if we consider the MP corresponding to Figure 5, the first ODE of Eq. (17) can be expressed as dC<sup>1</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>0</sup> <sup>V</sup><sup>1</sup> <sup>C</sup><sup>0</sup> <sup>þ</sup> <sup>Φ</sup><sup>21</sup> <sup>V</sup><sup>1</sup> <sup>C</sup><sup>2</sup> <sup>þ</sup> <sup>Φ</sup><sup>31</sup> <sup>V</sup><sup>1</sup> <sup>C</sup><sup>3</sup> � ð Þ <sup>Φ</sup>12þΦ<sup>13</sup> <sup>V</sup><sup>1</sup> C<sup>1</sup>

This equation—which obviously corresponds to an input tank—gives the first row of the MPmatrix (Eq. (18)) that can be written as � ð Þ <sup>Φ</sup><sup>12</sup> <sup>þ</sup> <sup>Φ</sup><sup>13</sup> V1 Φ<sup>21</sup> V<sup>1</sup> Φ<sup>31</sup> V1 .

The Gershgorin disk corresponding to this row is centered at <sup>a</sup><sup>11</sup> ¼ � ð Þ <sup>Φ</sup>12þΦ<sup>13</sup> <sup>V</sup><sup>1</sup> < 0 with radius <sup>R</sup><sup>1</sup> <sup>¼</sup> <sup>Φ</sup>21þΦ<sup>31</sup> <sup>V</sup><sup>1</sup> .

Now, if a flux balance is performed in this input tank, we have this equation: Φ<sup>0</sup> þ Φ<sup>21</sup> þ Φ<sup>31</sup> ¼ Φ<sup>12</sup> þ Φ13, and then Φ<sup>21</sup> þ Φ<sup>31</sup> < Φ<sup>12</sup> þ Φ<sup>13</sup> (at least if we consider the nontrivial case Φ<sup>0</sup> > 0). As a consequence of this fact, j j a<sup>11</sup> > R1, and the Gershgorin disk will look like the one schematized in Figure 6.

Now, if the second ODE of Eq. (17) is considered, this equation can be written as dC<sup>2</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>12</sup> <sup>V</sup><sup>2</sup> <sup>C</sup><sup>1</sup> <sup>þ</sup> <sup>Φ</sup><sup>32</sup> <sup>V</sup><sup>2</sup> <sup>C</sup><sup>3</sup> � ð Þ <sup>Φ</sup>21þΦ<sup>23</sup> <sup>V</sup><sup>2</sup> C2.

This internal tank equation corresponds to the second row of the MP-matrix (Eq. (18)): Φ<sup>12</sup> V2 � ð Þ <sup>Φ</sup><sup>21</sup> <sup>þ</sup> <sup>Φ</sup><sup>23</sup> V2 Φ<sup>32</sup> V2 .

Figure 6. The Gershgorin disk corresponding to an input tank.

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> <sup>V</sup><sup>2</sup> < 0 with radius <sup>R</sup><sup>2</sup> <sup>¼</sup> <sup>Φ</sup>12þΦ<sup>32</sup> <sup>V</sup><sup>2</sup> .

Since every eigenvalue lies within at least one of the Gershgorin disks, it follows that

As stated in Section 3, if there is no recirculation, then the ODE system has only negative

eigenvalues of the form , for all Then, in this case all the

In a previous work [6], it was proved that in an open MP, with three or less compartments, with or without recirculation, all the corresponding ODE system solutions are asymptotically stable. It is important to analyze if this result can be generalized or not, when closed systems and/or tanks with more than three compartments are considered. For this purpose, we will start with

In an open system, if the ith tank is an input one, then the diagonal entry of the ith row is aii < 0

If Φai, Φbi, ⋯, Φni are the incoming fluxes from other tanks of the system, ΦiA, ΦiB, ⋯, ΦiJ are the

<sup>0</sup> are the incoming fluxes from outside the system, then the

<sup>0</sup>C<sup>0</sup> <sup>þ</sup> … <sup>þ</sup> <sup>Φ</sup><sup>s</sup>

Square Matrices Associated to Mixing Problems ODE Systems

http://dx.doi.org/10.5772/intechopen.74437

53

<sup>0</sup>Cs (20)

Vi for k ¼ i, and

Cp (21)

P<sup>Φ</sup>ij

and aii j j > Ri being the sum of the non-diagonal entry modules of that row.

dt <sup>¼</sup> <sup>Φ</sup>aiCa <sup>þ</sup> … <sup>þ</sup> <sup>Φ</sup>niCn � <sup>Φ</sup>iA <sup>þ</sup> … <sup>þ</sup> <sup>Φ</sup>iJ � �Ci <sup>þ</sup> <sup>Φ</sup><sup>1</sup>

Φni Vi Cn � PΦij Vi

Ci þ

Vi ¼ aii j j, which proves the theorem.

PΦ<sup>p</sup> 0 Vi

Vi for k 6¼ i, �

<sup>0</sup> <sup>¼</sup> <sup>P</sup>Φij, which implies <sup>P</sup>Φki <sup>&</sup>lt; <sup>P</sup>Φij, and then:

Ca þ … þ

P<sup>Φ</sup>ij

Eq. (20) implies that the ith row of the MP-matrix has entries: <sup>Φ</sup>ki

P<sup>Φ</sup>ki Vi <

In the following section, these results—among others—will be generalized.

6. The general form of MP-matrices and new results

corresponding ODE system solutions will be asymptotically stable.

the following theorem.

outgoing fluxes, and Φ<sup>1</sup>

Vi dCi

This equation gives.

<sup>0</sup>, Φ<sup>2</sup>

dCi dt <sup>¼</sup> <sup>Φ</sup>ai Vi

Vi Cp is part of the independent term.

A flux balance gives <sup>P</sup>Φki <sup>þ</sup> <sup>P</sup>Φ<sup>p</sup>

Vi < 0 and also Ri ¼

corresponding ODE can be written as

<sup>0</sup>, ⋯, Φ<sup>s</sup>

Theorem 1

Proof

PΦ<sup>p</sup> 0

aii ¼ �

P<sup>Φ</sup>ij

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 schematized in Figure 7.

Finally, if the third ODE of Eq. (17) is considered, this equation can be written as dC<sup>3</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>13</sup> <sup>V</sup><sup>3</sup> <sup>C</sup><sup>1</sup> <sup>þ</sup> <sup>Φ</sup><sup>23</sup> <sup>V</sup><sup>3</sup> <sup>C</sup><sup>2</sup> � ð Þ <sup>Φ</sup>31þΦ32þΦ<sup>0</sup> <sup>V</sup><sup>3</sup> C3.

This output tank equation corresponds to the third row of the MP-matrix Eq. (18): Φ<sup>13</sup> V3 Φ<sup>23</sup> V3 � 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 .

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> <sup>V</sup><sup>3</sup> < 0 with radius <sup>R</sup><sup>3</sup> <sup>¼</sup> <sup>Φ</sup>13þΦ<sup>23</sup> <sup>V</sup><sup>3</sup> .

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

Taking into account all these results, the Gershgorin circles for the MP of Figure 5 are shown in Figure 8.

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

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

Since every eigenvalue lies within at least one of the Gershgorin disks, it follows that

In the following section, these results—among others—will be generalized.
