Positive Matrix Semigroups and Applications
Now we investigate positive one-parameter matrix semigroups, or, using a more common name, positive matrix exponentials. As expected, positivity and irreducibility in this case also lead to remarkable spectral and asymptotic properties.
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Positive Matrix Semigroups and Applications Now we investigate positive one-parameter matrix semigroups, or, using a more common name, positive matrix exponentials. As expected, positivity and irreducibility in this case also lead to remarkable spectral and asymptotic properties. Some applications of the theory are also presented to emphasize the importance of the subject.
7.1 Positive Semigroups In this section we combine the matrix exponential from Chapter 4 with the positivity from Chapter 5. More precisely, we consider positive matrix semigroups (etA )t≥0 , i.e., we assume that each etA , t ≥ 0, is a positive matrix. As a first step, we characterize this property by the entries of A. In particular, we show that positivity of A is sufficient, but not necessary for this. Let A = (αij ) be given. By Theorem 4.2, A = lim t↓0
etA − I , t
which can be rewritten coordinatewise as # tA $ e uj − uj , ui , αij = lim t↓0 t
(7.1)
for i, j = 1, . . . , n and ui the ith standard unit vector in Cn . If we denote the (i, j)th entry of etA by τij (t), then (7.1) implies that αij =
τij (t) t limt↓0 τii (t)−1 t
limt↓0
for i = j,
(7.2)
for i = j.
© Springer International Publishing AG 2017 A. Bátkai et al., Positive Operator Semigroups, Operator Theory: Advances and Applications 257, DOI 10.1007/978-3-319-42813-0_7
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Chapter 7. Positive Matrix Semigroups and Applications
If (etA )t≥0 is positive, i.e., τij (t) ≥ 0 for all t, i and j, then αij ≥ 0
for i = j, and
αii ∈ R
for i = j.
We call such matrices positive off-diagonal. Thus, we have shown the necessity part of the following characterization. Theorem 7.1. The matrix A = (αij ) ∈ L(X) generates a positive semigroup if and only if it is real and positive off-diagonal. Proof. It remains to show the sufficiency of the condition. Since A is real and positive off-diagonal, we can find ρ ∈ R such that Bρ := A + ρI ≥ 0
(7.3)
(e.g., take ρ := max1≤i≤n |αii |). Note that also etBρ ≥ 0 for all t ≥ 0. Applying the functional calculus introduced in Section 2.2 to the function f (λ) := etλ−tρ , we obtain etA = e[t(A+ρI)−tρI] = f (Bρ ) = e−tρ · etBρ ≥ 0 for all t ≥ 0.
Let us mention another terminology here. A real and positive off-diagonal matrix A is also called a Metzler matrix and −A is called a Z-matrix . Since etA ≥ 0 does not imply A ≥ 0, Perron’s theorem (see Theorem 5.6) is not directly applicable and r(A) may not be an eigenvalue of A. However, considering the positive matrix Bρ defined in (7.3) we obtain an important property of the spectral bound s(A) of A. Theorem 7.2. If A generates a positive semigroup (etA )t≥0 , then s(A) is a strictly dominant eigenvalue in the lateral sense, i.e., s(A) ∈ σ(A) and Re λ < s(A) for all other eigenvalues λ of A. Proof. As already noticed in the proof of Theorem 7.1, Bρ := A + ρI ≥ 0 for ρ := max1≤i≤n |αii |. Perron’s theorem (see Theorem 5.6) yields r(Bρ ) ∈ σ(Bρ ). Evidently, r(Bρ ) = s(Bρ ), which is strictly dominant in σ(Bρ ). Since σ(Bρ ) = σ(A + ρI) = σ(A) + ρ, and thus s(Bρ ) = s(A + ρI) = s(A) + ρ, we obtain that
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