The preceding section discussed several special cases of reducible matrices, but this was not our aim. Our aim is developing a method to find all nonnegative eigenvectors of a reducible matrix and to explain their behaviour. We saw that not the pattern of the eigenvalues was the most important factor, but the block structure of the matrix itself. The off-diagonal blocks determine the relationships that the classes/diagonal blocks have to each other. We can use these relationships to find out how the eigenvectors are build up. As before we shall illustrate the general idea by a convenient example, but first some additional definitions are introduced (See also Zijm, 1983, Ch.2).

Definition 1:

The term 'having access to' depends on the eigenvector which is taken into consideration.

left-hand eigenvector: A class j has access to class i iff A_ij is semipositive.

right-hand eigenvector: A class i has access to class j iff A_ij is semipositive.

Definition 2:

A class is said to be initial if no other class has access to it. (I).

A class is said to be final if it has no access to any other class (F).

A class which is neither initial nor final is called a transitclass (T).

A class which is both initial and final is called isolated (ISO).

Remark: The definition of an isolated class is rather different from the definition Gantmacher used.

Definition 3:

The set of all initial classes with respect to the left-hand/right-hand eigenvector is denoted by I_l/I_r.

The set of all final classes with respect to the left-hand/right-hand eigenvector is denoted by F_l/F_r.

The set of all transit classes with respect to the left-hand/right-hand eigenvector is denoted by T_l/T_r.

Remark: I_l is by definitionF_r, I_r is by definition F_l, and T_l is by definition T_r.

A class i belongs by definition to I_l iff A_ij=0 for all j. This is exactly the definition of F_r, so I_l is by definition F_r. The other proofs are analogous. Since T_l is by definitionT_r we shall denote, if necessary, the transit classes by T.

Definition 4:

If a class attains the dominant eigenvalue we call this class basic (B).

A class which does not attain the dominant eigenvalue is called nonbasic (NB).

Remark: It is obvious that the number of basic classes is equal to the algebraic multiplicity of λ.

Our convenient example will be the matrix

(17)

We can make two directed graphs of this matrix with the following rules corresponding to the left-hand and the right-hand eigenvector.

(18)

We shall use the graphs as a priority scheme to determine the equations that are most influential on the behaviour of the semipositive eigenvectors. Suppose that the classes 1,2,4,5, and 6 are basic classes. In that case we have the following graphs for both the eigenvectors.

Now take the right-hand eigenvector. Begin with the highest possible initial class (in this case class 8). This is the safest way, since a higher class can only point to lower classes. Class 8 is nonbasic. For a nonbasic class t in general the eigenvector equation can be written as

(19)

When A_tj is semipositive there will be an arc from class t to class j in the graph. When A_tj=0 then no arc is drawn. So the eigenvectorpart of a nonbasic class can be expressed in terms of the eigenvectorparts belonging to classes to which it is pointing directly in the graph. In this case the eigenvectorpart of class 8 can only be expressed in terms of the eigenvectorpart of class 7. Class 7 is nonbasic too, so it can be expressed in terms of the eigenvectorparts which belong to class 6 and class 4. Both classes are basic. What happens if a class t is basic? As before write w_t=b_ty_t and all the other w_j of the classes j to which class t is pointing are set to zero. In this example write w₆=b₆y₆, w₅=0 and w₂=0. Furthermore w₄=b₄y₄.

Here it is neccessary to take a little sidestep. Suppose that we did not begin with class 8, but with class 3 instead. Then we would have argued as follows. Class 3 is nonbasic, so w₃ can be expressed in terms of w₂. That can be written as w₃=(λ^*I-A₃₃)^-1A₃₂w₂. Since class 2 is basic set w₂ equal to b₂y₂ which yields

(20)

In this vector w₂ and w₃ are both positive. However, w₂ is known to be zero, and so is w₃, because w₃ still can be expressed in terms of w₂. The potential eigenvector in (20) therefore vanishes in order to avoid inconsisticies. This example shows that it makes sense to start the computation of the semipositive left-hand/ right-hand Perron eigenvector with the lowest/highest initial class. By doing otherwise one may easily be forced to revise the results.

Going back where we stopped, we see that since w₇ can be written in terms of w₆ and w₄. We have

(21)

w₈ can be expressed as

(22)

Together with the fact that class 1 is a basic isolated class, so w₁=b₁y₁, we can summarize our results as

(23)

We have found three semipositive eigenvectors by assuming that the b_i's are positive, but of course any linear combination of these vector belongs to the eigenspace. So, we can relax on this assumption and take one or more of the b_i negative if this suits our purposes.

We turn to left eigenvector in order to find the semipositive eigenvectors there. We start now with the lowest class, that is class 1. Since this is a isolated basic class we write v₁=a₁q₁. Now we turn to class 2. This is a basic class, and so v₂=a₂q₂, v₃=0 and v₆=0. Here we see what happens when an eigenvectorpart is zero. It sets the eigenvectorparts of the classes to which it is pointing equal to zero too! Thus v₆=0 implies v₇=0, which implies v₈=0. This leaves the initial classes 4 and 5. Both of them are basic so we write v₄=a₄q₄ and v₅=a₅q₅. Summarizing these results we find

(24)

Using this method it is easily seen that if the final classes are exactly the set of basic classes a positive eigenvector exists [Gantmacher, 1964][Zijm, 1983, ch.2]. In the next section it is indicated that the method finds all semipositive Perron eigenvectors (see also Cooper, 1973, for a formal graph theoretical approach).

MAIN