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ON THE ELASTICITY OF THE PERRON ROOT OF ANONNEGATIVE MATRIX∗
S. J. KIRKLAND† , M. NEUMANN‡ , N. ORMES‡ , AND J. XU‡
SIAM J. MATRIX ANAL. APPL. c© 2002 Society for Industrial and Applied MathematicsVol. 24, No. 2, pp. 454–464
Abstract. Let A = (ai,j) be an n × n nonnegative irreducible matrix whose Perron root is λ.
The quantity ei,j =ai,j
λ∂λ
∂ai,jis known as the elasticity of λ with respect to ai,j . In this paper,
we give two proofs of the fact that∂ei,j∂ai,j
≥ 0 so that ei,j is increasing as a function of ai,j . One
proof uses ideas from symbolic dynamics, while the other, which is matrix theoretic, also yields a
characterization of the case when∂ei,j∂ai,j
= 0. We discuss a resulting connection between the elements
of A and the elements of the group inverse of λI −A.
Key words. elasticity, population models, nonnegative matrices
AMS subject classifications. 15A09, 15A18, 15A48, 92D25
PII. S0895479801398244
1. Introduction. A large class of models in mathematical population biology(and in other areas) has the following common structure: we are given an n×n matrixA with nonnegative entries, frequently called the projection matrix of the model, andan initial population vector x0 ∈ R
n+, and, for each k ∈ N, we define xk = Axk−1. In
the case that the Perron root of A, say, λ, is a simple dominant eigenvalue, it followsthat xk/λ
k converges to an appropriate scalar multiple of the right Perron vector forA. Thus λ can be thought of as the asymptotic growth rate for the population beingmodeled.
A specific example and almost the simplest population model in mathematicalbiology is the Leslie model. In this model, individuals can live up to the age of n, andthe projection matrix of this model is given by
A =
F1 F2 . . . . . . Fn−1 Fn
T1 0 0 . . . 0 00 T2 0 0 0 0... . . .
. . .. . .
......
... . . . . . .. . .
......
0 . . . . . . . . . Tn−1 0
,(1.1)
where the Fi’s signify the birth rate (or fecundity) at age i, i = 1, . . . , n, while theTi’s signify the survival rate from age i to age i+1, i = 1, . . . , n− 1. In the literatureon mathematical models for population growth, the birth and survival rates togetherare often referred to as the vital rates. We refer the reader to Caswell [5] for acomprehensive introduction and a reference source on matrix population models.
∗Received by the editors November 13, 2001; accepted for publication (in revised form) by R.Bhatia April 29, 2002; published electronically November 6, 2002.
http://www.siam.org/journals/simax/24-2/39824.html†Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada
S4S 0A2 ([email protected]). The research of this author was supported in part by NSERCunder grant OGP0138251.
‡Department of Mathematics, University of Connecticut, Storrs, CT 06269–3009 ([email protected], [email protected], [email protected]). The work of the second author wassupported in part by NSF grant DMS9973247.
454
ELASTICITY OF PERRON ROOT 455
While the perturbation analysis of the Perron root λ of a nonnegative and ir-reducible matrix A occurs in a variety of applications of nonnegative matrices (see,for example, Berman and Plemmons [3] and Varga [13], where further backgroundmaterial on nonnegative and M-matrices can also be found), the usual sensitivityanalysis neglects inherent restrictions on the magnitudes of the various entries of A.Returning to the example of the Leslie population model given above, an entry of Ain (1.1) corresponding to a birth rate may exceed 1, while an entry corresponding toa survival rate must necessarily be at most 1. Consequently, in order to take thesemagnitude restrictions into account, the notion of the elasticity of λ with respect toan entry or vital rate in A has been introduced in the mathematical biology literature(see [5] and De Kroon, Plaisier, van Groenendael, and Caswell [6]). Here is its formaldefinition.
Definition 1.1. Let A = (ai,j) be a nonnegative matrix, and suppose that itsPerron root λ is a simple eigenvalue. The elasticity of λ with respect to the (i, j)thentry of A is given by
ei,j =ai,jλ
∂λ
∂ai,j, i, j = 1, . . . , n.(1.2)
If we regard ∂λ/∂ai,j as the measure of the sensitivity of λ to a change in ai,j ,then we can view the elasticity with respect to the (i, j)th entry as the proportionalsensitivity of λ to a change in ai,j . We note that, from (1.2), ei,j also admits therepresentation as
ei,j =∂ log λ
∂ log ai,j, i, j = 1, . . . , n.
Thus the elasticity can be thought of as measuring the multiplicative change in λ dueto a multiplicative change in ai,j , while the sensitivity measures the additive effecton λ arising from an additive change in ai,j . Finally, we note that, in [6], it is shownthat
n∑i,j=1
ei,j = 1
so thatn∑
i,j=1
ei,jλ = λ.
In this way, the elasticities ei,j provide a quantification of the contribution of ai,j tothe size of λ.
Throughout this paper, we will focus on the fundamental case that A is irre-ducible. This case is of both practical and theoretical interest, and it is well knownthat, in this case, the Perron root of A is simple so that, for each entry in A, thecorresponding elasticity is well defined.
In [5, 9.7.1], Caswell discusses the sensitivity of the elasticities to changes in thevital rates and deduces from (1.2) that
∂ei,j∂ak,�
=ai,jλ
∂2λ
∂ai,j∂ak,�− ai,j
λ2
∂λ
∂ak,�
∂λ
∂ai,j(1.3)
+δi,kδj,�
λ
∂λ
∂ai,j, i, j, k, � = 1, . . . , n,
456 S. J. KIRKLAND, M. NEUMANN, N. ORMES, AND J. XU
where δp,q is 1 or 0 according to whether p = q.Fortunately, formulae are available for the various partial derivatives that appear
in (1.3). For an n× n nonnegative and irreducible matrix A with Perron root λ andright and left Perron vectors x and wT , respectively, normalized so that wTx = 1, itis well known that
∂λ
∂ai,j= wixj , i, j = 1, . . . , n(1.4)
(see Wilkinson [15] or Stewart [12], for example). Further, in [7], it is shown that
∂2λ
∂ai,j∂ak,�=
∂λ
∂ai,�Q#
j,k +∂λ
∂ak,jQ#
�,i, i, j, k, � = 1, . . . , n,(1.5)
where Q# is the group inverse of the singular irreducible M-matrix Q = λI −A. (SeeBen-Israel and Greville [2] and Campbell and Meyer [4] for background material ongeneralized inverses.) Substituting (1.4) and (1.5) into (1.3), we see that, in particular,
∂ei,j∂ai,j
=1
λwixj
(2ai,jQ
#j,i − ai,j
1
λwixj + 1
).(1.6)
Thus we find that the elasticity of the Perron root with respect to the (i, j)th entryis increasing as a function of the (i, j)th entry if and only if
2ai,jQ#j,i − ai,j
1
λwixj + 1 ≥ 0.(1.7)
In this paper, we give two different proofs of the fact that, for each pair of indicesi, j, the quantity ei,j is increasing as a function of ai,j . The first proof, developedin section 2, is matrix theoretic, and it also yields a characterization of the case ofequality in (1.7). The second proof, developed in section 3, relies on techniques fromthe theory of symbolic dynamics. In section 4, we give some closing remarks. Forconvenience, we now state our main result.
Theorem 1.2. Let A = (ai,j) be an irreducible nonnegative matrix of order n withPerron root λ and right Perron vector x = (x1, . . . , xn)
T . Then, for each 1 ≤ i, j ≤ n,ei,j is an increasing function of ai,j. Specifically,
∂ei,j∂ai,j
≥ 0.(1.8)
Moreover,
∂ei,j∂ai,j
= 0(1.9)
if and only if A is permutationally similar to the matrix λDAD−1, where D is thediagonal matrix whose ith diagonal entry is xi, i = 1, . . . , n, and where stochasticmatrix A is periodic and has the form
A =
0 1 0 0 · · · 00 0 X1 0 · · · 00 0 0 X2 · · · 0...
. . ....
0 0 0 · · · 0 Xp
1 0 0 · · · 0 0
,(1.10)
where 1 denotes the all-ones vector, where the ith row of A corresponds to the firstrow of A, and where X1 has just one row, which corresponds to row j of A.
ELASTICITY OF PERRON ROOT 457
2. A matrix theoretic proof of Theorem 1.2. In this section, we give amatrix theoretic proof of all of the conclusions of Theorem 1.2.
Let A = (ai,j) be an n × n irreducible nonnegative matrix whose Perron root isλ. If x = (x1, . . . , xn)
T is a right Perron vector of A and D is the diagonal matrixwhose ith diagonal entry is xi, i = 1, . . . , n, then it is well known (see [3, Theorem2.5.4]) that the matrix A := 1
λD−1AD is stochastic. Let Q = λI −A and Q = I − A.
Then Q# = λD−1Q#D, and it is not difficult to show, from (1.6), that, if E = (ei,j)
and E = (ei,j) are the matrices of elasticities arising from A and A, respectively, then
∂ei,j∂ai,j
=1
λ
xj
xi
∂ei,j∂ai,j
, i, j = 1, . . . , n.
We thus conclude that
sign
(∂ei,j∂ai,j
)= sign
(∂ei,j∂ai,j
), i, j = 1, . . . , n,
and so, for our purposes in this section, it will suffice to consider the case that ouroriginal n× n irreducible nonnegative matrix A is stochastic.
We begin with the following lemma.Lemma 2.1. Let B be a substochastic matrix of order n ≥ 2 whose spectral radius
is less than 1. Fix an index j, with 1 ≤ j ≤ n, and, for each l ∈ N, let αl = eTj Bl1.Then
∞∑l=1
α2l + 2
∞∑l=1
∞∑m=l+1
αlαm ≤∞∑l=1
αl + 2
∞∑l=1
(l − 1)αl.(2.1)
Suppose further that each vertex in the digraph of B can be reached from j by somewalk. Then, if equality holds in (2.1), there is a p ∈ N such that B can be permutedto the form
0 X1 0 0 · · · 00 0 X2 0 · · · 0...
. . ....
0 0 0 · · · 0 Xp
0 0 0 · · · 0 0
,
where Xi1 = 1 for i = 1, . . . , p, and where X1 has only one row, which correspondsto index j.
Proof. For each l ∈ N, 0 ≤ αl ≤ 1, so we see that∑∞
l=1 α2l ≤∑∞
l=1 αl. Also,
∞∑l=1
∞∑m=l+1
αlαm =
∞∑m=2
m−1∑l=1
αlαm ≤∞∑
m=2
(m− 1)αm.
The inequality (2.1) now follows readily.Suppose now that equality holds in (2.1). Then, in particular, we must have that
αl = α2l for each l so that αl is either 1 or 0 for each l. Note that, since Bl → 0 as
l → ∞, we see that αp = 0 for some p. However, that implies that, in the digraphof B, there is no walk of length p starting from vertex j and hence no walk of lengthlonger than p starting from j. (Note, in particular, that the digraph has no cycles.)We conclude that, for some p, we have αl = 1 if l ≤ p and αl = 0 if l ≥ p+ 1.
458 S. J. KIRKLAND, M. NEUMANN, N. ORMES, AND J. XU
We claim that this last condition implies that the vertices in the digraph of Bwhich are distinct from j can be partitioned into sets S1, . . . , Sp such that, for eachi, Si is the set of vertices v such that the distance from j to v is i. We prove theclaim by induction and note that, for the case when p = 1, each vertex distinctfrom j must be in the outset of j, giving the desired partitioning. Next, supposethat the claim holds for some p ≥ 1 and that we have that αl = 1 if l ≤ p + 1and αl = 0 if l ≥ p + 2. Let S1 be the outset of j, and note that, for each l ≥ 2,αl =
∑ni=1 bj,iαi,l−1, where αi,l−1 = eTi Bl−11. It follows that αi,l = 1 for 1 ≤ l ≤ p
and αi,l = 0 for l ≥ p + 1. Thus, for each vertex i ∈ S1, the induction hypothesisapplies to those vertices reachable from i, yielding a corresponding partitioning of thevertex set. However, a vertex at a distance d from i is necessarily at a distance d+ 1from j, and the desired partitioning follows, completing the induction step.
From the above claim, it now follows that we can write B in the form
0 X1 0 0 · · · 00 0 X2 0 · · · 0...
. . ....
0 0 0 · · · 0 Xp
0 0 0 · · · 0 0
.
Finally, the fact that Xi1 = 1 for i = 1, . . . , p now follows since αl = 1 for 1 ≤ l≤ p.
Lemma 2.1 yields the following corollary.
Corollary 2.2. Let B be as in Lemma 2.1, and fix an index j. Then
eTj (I −B)−11+[eTj (I −B)−11
]2 ≤ 2eTj (I −B)−21.(2.2)
Suppose also that each vertex in the digraph of B can be reached from j by some walk.If equality holds in (2.2), then there is a p ∈ N such that B can be written as
0 X1 0 0 · · · 00 0 X2 0 · · · 0...
. . ....
0 0 0 · · · 0 Xp
0 0 0 · · · 0 0
,
where Xi1 = 1 for i = 1, . . . , p and where X1 has only one row, which corresponds toindex j.
Proof. As in Lemma 2.1, we let αl = eTj Bl1 for each l ∈ N. Note that (I−B)−1 =∑∞l=0 Bl and that (I − B)−2 =
∑∞l=1 lBl−1. Thus we see that 2eTj (I − B)−21 =
2 + 2∑∞
l=2 lαl−1, while eTj (I −B)−11 = 1 +∑∞
l=1 αl. Consequently, the inequality
2eTj (I −B)−21 ≥ eTj (I −B)−11+[eTj (I −B)−11
]2is equivalent to the inequality
2 + 2∞∑l=2
lαl−1 ≥ 1 +
∞∑l=1
αl +
(1 +
∞∑l=1
αl
)2
.
ELASTICITY OF PERRON ROOT 459
However, this last inequality is easily seen to simplify to
∞∑l=1
αl + 2
∞∑l=1
(l − 1)αl ≥∞∑l=1
α2l + 2
∞∑l=1
∞∑m=l+1
αlαm.
The results, including the equality case, now follow from Lemma 2.1.Consider the case when i = j in (1.7) for a stochastic matrix A with left Perron
vector wT , normalized so that its entries sum to 1. In that situation, the left side of(1.7) becomes 2ai,iQ
#i,i−ai,iwi+1. It is shown in [7] thatQ
#i,i > 0 for each i = 1, . . . , n,
and it follows readily then that∂ei,i∂ai,i
≥ wi(1− ai,iwi) > 0. Thus, in order to establish
Theorem 1.2, we need only consider the case when i = j. That case is (essentially)considered in the following proposition.
Proposition 2.3. Let A be an irreducible stochastic matrix of order n ≥ 3,written as
A =
m0 m1 · · · mn−1
y B
.
Then, for each 1 ≤ j ≤ n− 1,
∂e1,j+1
∂a1,j+1≥ 0.(2.3)
Furthermore,
∂e1,j+1
∂a1,j+1= 0(2.4)
if and only if A is permutationally similar to a matrix A of the form
A =
0 1 0 0 · · · 00 0 X1 0 · · · 00 0 0 X2 · · · 0...
. . ....
0 0 0 · · · 0 Xp
1 0 0 · · · 0 0
,(2.5)
where the first row of A corresponds to the first row of A and where X1 has just onerow, which corresponds to row j + 1 of A.
Proof. Let mT = (m1, . . . ,mn−1) ∈ R1,n−1, and let wT be the left Perron vector
for A whose entries sum to 1. Note that w1 = 1/[1 + mT (I − B)−11]. Also, ifQ = I −A, then, for j = 1, . . . , n− 1, we have, using Meyer [10, (5.1)] in conjunctionwith a permutation similarity, that
Q#1,j+1 = w2
1mT (I −B)−21− w1e
Tj (I −B)−11.
It now follows, from (1.6), that
1
w31
∂e1,j+1
∂a1,j+1= 2mjm
T (I −B)−21
− 2mjeTj (I −B)−11
[1 +mT (I −B)−11
]−mj
[1 +mT (I −B)−11
]+[1 +mT (I −B)−11
]2 ≡ f.
(2.6)
460 S. J. KIRKLAND, M. NEUMANN, N. ORMES, AND J. XU
Suppose that i = 0, j, and note that
∂f
∂mi= mj
[2eTi (I −B)−21− eTi (I −B)−11
]+ 2eTi (I −B)−11
[1 +mT (I −B)−11−mje
Tj (I −B)−11
] ≥ 0.
Thus, in order to show that f is nonnegative, it suffices to show that fact in the casewhen, for some t ∈ [0, 1], mj = t and m0 = 1−t. However, in that instance, f reducesto
1− t+ t2[2eTj (I −B)−21− eTj (I −B)−11− (eTj (I −B)−11)2
].
Appealing to Corollary 2.2, we see that f ≥ 0.Next suppose that f = 0. We find, from the above, that necessarily mj = 1 and
2eTj (I −B)−21 = eTj (I −B)−11+[eTj (I −B)−112
].
Since A is irreducible, the spectral radius of B is less than 1. Also, since each vertexin the digraph of A can be reached by a walk starting from vertex 1 and since eachwalk starting from 1 must pass immediately through j + 1, we see that each vertexdistinct from 1 and j + 1 is reachable by a walk from j + 1 which is contained in thedigraph of B. Thus we see that all of the hypotheses of Corollary 2.2 apply to B. Asa result, there is a p such that we may write B as
0 X1 0 0 · · · 00 0 X2 0 · · · 0...
. . ....
0 0 0 · · · 0 Xp
0 0 0 · · · 0 0
,
where Xi1 = 1 for i = 1, . . . , p and where the X1 has only one row which correspondsto index j + 1. Using the irreducibility and stochasticity of A, we deduce that A ispermutationally similar to a matrix A having the form of (2.5).
Finally, if A is permutationally similar to the matrix A of (2.5), then it follows
from results in [9] that Q#j+1,1 = −(p+ 1)/(2p+ 4), while w1 = 1/(p+ 2) so that, by
(1.6),
∂e1,j+1
∂a1,j+1= 0.
3. A proof of (1.8) via symbolic dynamics. Our second proof of (1.8) re-lies on a well-known principle from the theory of symbolic dynamical systems, thevariational principle for pressure. An adapted form of that principle is given belowand is stated in the language of nonnegative matrices. A more authentic form of thisprinciple can be found in Walters [14] and in Arnold, Gundlach, and Demetrius [1].
Theorem 3.1 (variational principle for pressure, restated). Let A = (ai,j) be ann× n irreducible nonnegative matrix with Perron root λA and right Perron vector x.Let MA be the collection of all n× n stochastic nonnegative matrices P = (pi,j) suchthat ai,j = 0 ⇔ pi,j = 0. For each P ∈ MA, let rP be the left Perron vector of Pwhose entries sum to 1. Then
log λA = supP∈MA
−
n∑i,j=1
(rP )ipi,j log pi,j +
n∑i,j=1
(rP )ipi,j log ai,j
.
ELASTICITY OF PERRON ROOT 461
Furthermore, the supremum is achieved at the stochastic matrix PA such that, foreach 1 ≤ i, j ≤ n,
(PA)i,j =ai,jxj
λAxi.(3.1)
In our next result, we apply the characterization of the Perron root given inTheorem 3.1 in order to describe the elasticity with respect to a particular entry ofA.
Proposition 3.2. The elasticity ek,� evaluated at the matrix A = (ai,j) is equalto (rPA
)k(PA)k,�.Proof. Let w and x be positive vectors satisfying Ax = λAx and wTA = λAwT ,
and note that
ek,� =ak,�λA
wkx�
wTx.
Notice also that rTPA= (1/wTx)(x1w1, x2w2, . . . , xnwn) since, for each 1 ≤ j ≤ n, we
have
n∑i=1
1
wTxxiwi(PA)i,j =
n∑i=1
1
wTxxiwi
ai,jxj
λAxi
=
n∑i=1
1
wTx
wiai,jxj
λA
=1
wTxxjwj ,
(3.2)
while clearly ∥∥∥∥ 1
wTx(x1w1, x2w2, . . . , xnwn)
∥∥∥∥1
= 1.
We thus conclude that
(rPA)k(PA)k,� =
1
wTxwkxk
ak,�x�
λAxk=
ak,�λA
wkx�
wTx= ek,�.
With Theorem 3.1 and Proposition 3.2 in mind, fix an ordered pair (k, �), 1 ≤k, � ≤ n, and let B = (bi,j) be a nonnegative matrix whose entries are as follows:bi,j = ai,j for all (i, j) = (k, �) and bk,� > ak,�. Denoting ek,� evaluated at A and atB by ek,�|A and ek,�|B , respectively, we see that, if ak,� = 0, then ek,�|A = 0 whileek,�|B > 0 so that ek,�|B > ek,�|A. Thus, if ak,� = 0, then ek,� is increasing in ak,�.
Next assume that ak,� > 0, and let λB be the Perron root of B. Since B ≥ A,but A and B differ only in the (k, �) position (where each has a positive entry), wesee that MA =MB . Then, by Theorem 3.1,
log λB ≥ −∑i,j
(rPA)i(PA)i,j log(PA)i,j +
∑i,j
(rPA)i(PA)i,j log bi,j
= log λA + (rPA)k(PA)k,� [log bk,� − log ak,�] .
(3.3)
From (3.3) and Proposition 3.2, we see that
log λB − log λA
log bk,� − log ak,�≥ ek,�|A.(3.4)
462 S. J. KIRKLAND, M. NEUMANN, N. ORMES, AND J. XU
Similarly, we also find from Theorem 3.1 that
log λA ≥ −∑i,j
(rPB)i(PB)i,j log(PB)i,j +
∑i,j
(rPB)i(PB)i,j log ai,j
= log λB + (rPB)k(PB)k,� [log ak,� − log bk,�] .
(3.5)
Applying Proposition 3.2 to the matrix B, it follows from (3.5) that
ek,�|B ≥ log λB − log λA
log bk,� − log ak,�.(3.6)
Consequently, from (3.4) and (3.6), we find that
ek,�|B ≥ ek,�|Aso that ek,� is nondecreasing in the (k, �) entry of A. In particular, (1.8) followsreadily.
4. Examples and remarks. We begin this section with an example illustratingthe case of equality in Theorem 1.2. Let
A =
0 1 0 0 0 0
0 0 1 1 0 0
0 0 0 0 1 1
0 0 0 0 1 1
1 0 0 0 0 0
1 0 0 0 0 0
.
Then calculations show that λ =√2, and the matrix of elasticities is given by
E = (ei,j) =
0 1/4 0 0 0 0
0 0 1/8 1/8 0 0
0 0 0 0 1/16 1/16
0 0 0 0 1/16 1/16
1/8 0 0 0 0 0
1/8 0 0 0 0 0
.
From Theorem 1.2, we anticipate that ∂ei,j/∂ai,j is 0 only for i = 1 and j = 2, whilethe remaining quantities are positive. This is indeed the case; our computations yield
(∂ei,j∂ai,j
)=
0.17678 0 0.17678 0.17678 0.12500 0.12500
0.12500 0.17678 0.062500 0.062500 0.088388 0.088388
0.088388 0.12500 0.088388 0.088388 0.046875 0.046875
0.088388 0.12500 0.088388 0.088388 0.046875 0.046875
0.062500 0.17678 0.12500 0.12500 0.088388 0.088388
0.062500 0.17678 0.12500 0.12500 0.088388 0.088388
.
ELASTICITY OF PERRON ROOT 463
Theorem 1.2 shows that, for an n× n nonnegative irreducible matrix A = (ai,j),∂ei,j/∂ai,j is bounded below by 0; the following example shows that these derivativesare not bounded from above. Let J be the n×n all-ones matrix, let α ∈ (0, 1), and letA = (α/n)J so that the Perron root of A is α. Let Q = αI−A. It follows readily that
Q# = (1/α2)Q. In this case, ai,i = α/n, Q#i,i = (1/α)(1 − 1/n), and ∂λ/∂ai,i = 1/n
for all 1 ≤ i ≤ n. Substituting these three expressions in (1.6), we obtain that, for all1 ≤ i ≤ n,
∂ei,i∂ai,i
=n2 + 2n− 3
αn3
and, similarly, that, for distinct indices i, j with 1 ≤ i, j ≤ n, we have
∂ei,j∂ai,j
=n2 − 3
αn3.
Observe that each of these quantities can be made arbitrarily large by choosing thepositive parameter α sufficiently close to 0.
We close with a consequence of (1.8). Suppose that we have an irreducible stochas-tic matrix A of order n, and let wT denote its left Perron vector, normalized so thatits entries sum to 1. (In particular, A can be thought of as the transition matrix ofa Markov chain with stationary distribution vector w.) Letting Q = I − A, it turnsout that the modulii of the entries in Q# can be used to measure the stability of thecomputation of wT . Specifically, Funderlic and Meyer [8] propose maxi,j=1,... ,n |Q#
i,j |as a condition number for the Markov chain, while Meyer [11] suggests ||Q#||∞ asa condition number for the chain. From (1.8), we find that, for each pair of indices1 ≤ i, j ≤ n,
2ai,jQ#j,i − ai,jwi + 1 ≥ 0.(4.1)
Since wTQ# = 0T and Q#1 = 0 and since the diagonal entries of Q# are positive,it follows that Q# has at least one negative entry in each row and column. Supposenow that Q#
j,i < 0 and that ai,j > 0. Then, from (4.1), we see that
|Q#j,i| = −Q#
j,i ≤ 1− ai,jwi
2ai,j<
1
2ai,j.(4.2)
Thus we find that (1.8) can be used to provide an upper bound on the modulii of someof the negative entries of Q# in terms of the entries in A and wT . This observationmay be useful in discussing the condition numbers mentioned above.
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