Indecomposability and Primitivity of Nonnegative Matrices
| Autor | Takao Fujimoto; Fumiko Ekuni |
| Cargo | University of Kagawa, Japan/Shikoku-Gakuin Junior College, Japan |
| Páginas | 163-176 |
Page 163
In Leontief models, the concepts of indecomposability and primitivity play an important role with the former meaning the interconnectivity among industries and the latter meaning dynamic stability of given models. In this paper, we restate a proposition due to Frobenius, which has not been presented in the textbooks on linear algebra. We also explain about two applications of this proposition to population dynamics by Leslie and to social networks. In Section 2, we define indecomposability and primitivity of nonnegative matrices, and give our main proposition on a characterization of primitivity in terms of indecomposability. Section 3 is devoted to Leslie model of population dynamics, and Section 4 to the concepts of degree of connection and fragility of social networks. The last Section 5 includes some remarks.
We first explain our notation. The symbol Rn means the Euclidean space of dimension n(n R ≥ 2), R+n is the nonnegative orthant of Rn, and A is a givenPage 164 n x n nonnegative matrix. Let aij be the (i, j)-element of the matrix A, and xj the j-th entry of the vector x ∈ Rn. The symbol N stands for the index set N ≡ {1, 2, ..., n}. For vector comparison, we use the following inequality signs.
[SEE FORMULAS IN THE ATTACHED PDF]
Now we define
Definition 2.1.
[SEE TEXT IN THE ATTACHED PDF]1
A nonnegative matrix A is indecomposable iff it is not decomposable. Equivalently, however, we may define the indecomposability as
Definition 2.2.
[SEE TEXT IN THE ATTACHED PDF]
Definition 2.3.
[SEE TEXT IN THE ATTACHED PDF]2
It is well known that a nonnegative matrix which is indecomposable and has at least one positive diagonal element is primitive. This is obvious from another well-known characterization of indecomposable imprimitive matrices explained, e.g., in Nikaido (1963, Theorem 8.2, p. 117) or Nikaido (1970, Theorem 21.1, p. 142).3 This proposition dates back to G. Frobenius.4 A direct andPage 165 combinatorial proof is given in a forthcoming book by Krause, even in a nonlinear setting.5 In this paper, a direct and simple proof is presented, which can be used also in nonlinear generalizations.
Let us consider the following 2 × 2 matrix A
[SEE THE FORMULA IN THE ATTACHED PDF]
This is indecomposable. When we make the second power of A,
[SEE THE FORMULA IN THE ATTACHED PDF]
which is decomposable. Thus, we need an additional condition to guarantee the indecomposability of the power of an indecomposable matrix.
We first state a lemma.
Lemma 2.1. If A is indecomposable and has at least one positive diagonal element, then Ak is also indecomposable for any integer k > 0.
Proof. We prove this lemma using mathematical induction. First for k = 1, the proposition is obvious. We assume this lemma holds for (k – 1) with k ≥ 2. Let us also assume without losing generality that a11 > 0.
[SEE TEXT IN THE ATTACHED PDF]
Page 166
[SEE TEXT IN THE ATTACHED PDF]
Now we prove our main proposition.
Proposition 2.2. A given nonnegative matrix A is primitive if and only if Ak is indecomposable for any positive integer k.
[SEE TEXT IN THE ATTACHED PDF]
Proof.
[SEE TEXT IN THE ATTACHED PDF]
Corollary 2.3. If a given indecomposable matrix has its trace positive, then it is primitive.
Proof. It is obvious from Lemma 2.1 and Proposition 2.2. QED.
Page 167
We are ready to apply our results in the previous section to Leslie’s model of population dynamics, which is a discrete version of Lotka’s model.6 This model consists of n age groups and it is represented by a nonnegative n × n (≥ 2) matrix A, which shows the survival ratios of age groups after one period, let us say an ‘year’, and reproduction ratios (or birth rates) for some age groups. (We assume away the differences between males and females.) More specifically, the matrix A is written as
[SEE TEXT IN THE ATTACHED PDF]
Page 168
Assumption 3.1.
[SEE TEXT AND FORMULAS IN THE ATTACHED PDF]
this is indecomposable. This means that if there are at least two age groups which are reproductive, the matrix Ay is primitive. To prove this, consider
[SEE TEXT AND FORMULAS IN THE ATTACHED PDF]
Page 169
[SEE TEXT IN THE ATTACHED PDF]
Proposition 3.1.
[SEE TEXT IN THE ATTACHED PDF]78
In this subsection, we raise a numerical example of Leslie model. Let there be six age groups in our society, and the survival-birth ratio matrix is given as follows:
[SEE THE FORMULA IN THE ATTACHED PDF]
With this matrix, we have
[SEE FORMULAS IN THE ATTACHED PDF]
Page 170
In fact, among eigenvalues of A, there exists a unique positive eigenvalue, λ, called the Frobenius root of A, with which a strictly positive eigenvector is associated:
λ = 1.05, and its eigenvector x= (0.04 0.24 0.35 0.39 0.42 0.74)’.
Starting from an arbitrary initial vector of distribution, the relative age distribution gets near the above eigenvector, and the rate of...
Para continuar leyendo
Solicita tu prueba