Compartmental matrices with single root and nonnegative nilpotent matrices

MATHEMATICAL

Compartmental Nilpotent

BIOSCIENCES

135

14, 135-142 (1972)

Matrices with Single Root and Nonnegative

Matrices

JOHN 2. HEARON Mathematical Research Branch, National Institute of Arthritis and Metabolic Building 31, Room 9AllA, National Institutes of Health, Bethesda, Maryland

Diseases, 20014

ABSTRACT It is shown that, as a consequence of the Frobenius Theorem on nonnegative matrices, an n-square compartmental matrix with a single n-fold root is permutationally similar to a triangular matrix. As a result, certain properties of precedence matrices and some graph theoretic results can be viewed as essentially resulting from the classical theorem on nonnegative irreducible matrices.

INTRODUCTION Let x(t) be the amount of material at time t in the nth compartment of an n-compartment system for which the matrix is nonsingular and upper Hessenberg (zero entries below the subdiagonal), and let T be that (unique) value of t at which x attains its maximum. It is shown in [1] that TX(T)/ mx(t)at < (n -

l)“[exp(l

- ~z)]/(n -

I)! and that strict equality

obtains

if and only if the roots of the matrix satisfy 2, = 2, = . . . = i,,. Now this condition is realized in the familiar system X1 --) X2 -+ . . . -+ X,, -f: provided ki is the rate constant for the transfer of material from the ith compartment and k, = k2 = . . . = k,,. Given then that the set of all systems for which the condition can be realized is not empty, we seek to characterize such systems. It is shown here that the set of all upper Hessenberg compartmental matrices of order n which have a single n-fold root is precisely the intersection of the set of all upper Hessenberg compartmental matrices of order n and the set of all lower triangular matrices of order n. This result is obtained as a corollary of the theorem, proved in what follows, that a real matrix of order n with nonnegative off-diagonal entries has a single n-fold root only if it is permutationally similar to a triangular matrix. A very special case of this theorem is that 3.r = Eu2= . . . = I.,, = 0, in which case the matrix is nilpotent. Thus we obtain a characterization of Copyright 0 1972 by American Elsevier Publishing Company, Inc.

136

JOHN Z. HEARON

nonnegative nilpotent matrices. This enables us to view certain known properties of precedence matrices and graph theoretic results as stemming essentially from the classica! Frobenius theorem regarding irreducible nonnegative matrices. SOME

PRELIMINARIES

We call a matrix upper Hessenberg if all entries below the subdiagonal are zero: aij = 0 for i > j + 2. We call a matrix lower triangular if all entries above the main diagonal are zero: aij = 0 forj 3 i + 1. In certain applications an important case of upper Hessenberg is the continuant or tridiagonal matrix [l-3]. A matrix which is both upper Hessenberg and lower triangular has nonzero entries only on the main diagonal and subdiagonal. Given any square matrix A, of order II, we construct the corresponding directed, labeled graph G(A), henceforth simply called the graph of A, as follows: we begin with n points or vertices ul, u2, . . ., u,. If aij # 0 we draw a directed line (arrow) from j to i. In particular if i = j, we draw a loop or sling originating in vi and terminating in vi. If for some set of (not necessarily distinct) indices iI, i2, . . ., iS_I the sequence of vertices and arrows, vi + vi, + vi2 + . . . vi,_1 -+ vi, exists in G(A) we say there is a sequence of length s connecting Uj to Vi, In this case no element of the sequence aii,_,, is zero. Conversely if no element in such a sequence vanishes then there is a sequence of length s connecting vj to vi. If the vertices of the sequence are distinct we call the sequence a path of length s. Obviously every sequence contains a path. . .

.,

CZi2ilr

CZifj

Remark The term path as used here agrees with [5]. Some authors [6, 71 call a sequence in which no arrow is repeated (but in which a given vertex may be traversed more than once) a path and define a simple path as the object which is here called a path. A cycle of order s is obtained from a path of length s by adding the arrow connecting the terminal vertex to the initial vertex. We regard a loop or sling vi -+ vi (corresponding to aii # 0) as a cycle of order one. Following Goldberg [4], we define a simple loop as a cycle vi --f vi, + vi2 + . . . vi,_, + vi such that the vertices vi, vi,, . . ., vi,_1 have no other arrows between them. Observe that a cycle of order two is of necessity a simple loop. It is clear that some subset of the vertices in a cycle lie in a simple loop. For, let v;, vi, . . ., v; lie in a cycle of order q. Assume that the cycle is not a simple loop and that no pair of vertices lie in a cycle of order two, since in the contrary cases there is nothing to

MATRICES

WITH SINGLE ROOT

137

discuss. Then some v; is connected to v; where i # ,j + 1. Ifj + 1 < i d IZ then there is a cycle of order < q from vl, to v\ and ifj - 1 > i 3 1 then there is a cycle of order < q from v: to vi. In either case we consider this shorter cycle and, if it is not a simple loop, repeat the argument until we arrive at a simple loop. This cannot fail, for if nothing else we arrive at a cycle of order three which must be a simple loop. The adjacency matrix [5] (vertex incidence matrix [6], vertex matrix [7]) of G(A) is the matrix M,(G) formed by entering 1 in the (i,j)-position if there is an arrow from vi to vi in G(A); M,(G) also results from A if each aij # 0 is replaced by 1. It is obvious that the adjacency matrix of a simple loop is a permutation matrix (exactly one 1 in each row and each column) amd that corresponding to a simple loop in G(A) is a principal submatrix of M,,(G) which is a permutation matrix. Any permutation of the columns of A followed by the same permutation of rows produces a permutationally similar matrix: B = PAP-‘, where P is a permutation matrix. Clearly G(B) is obtained from G(A) by renumbering the vertices of G(A). Conversely, any relabeling of G(A) results in a graph which is the graph of B = PAP-’ for appropriate permutation matrix P. COMPARTMENTAL

MATRICES

WITH

SINGLE

ROOT

The results of this section were motivated by compartmental analysis; and the main applications, other than those of the next section, are in linear kinetics [I]. However we deal here with a slightly broader class of matrices, viz., those real, square matrices A such that aij 2 0 for i # j. Given this, it is the character of the aii which determines if there is a realizable compartmental system corresponding to the matrix A. It is well known that (e.g. see [8, 91) that it is both necessary and enough to have aii < 0 and diagonal dominance. There will be dominance with respect to columns (rows) if the independent variable is quantity of tracer (specific activity) [81. THEOREM

1

Let A be real with nonnegatioe qfS-diagonal entries and roots 2,. i.,, . . ., A,,. Then I., = 2, = . . . = A, if and only if A is permutationally similar to a triangular matrix and a,, = az2 = . . . = a ,,“. Proqf If A is similar to a triangular matrix then the roots coincide with the diagonal entries. Conversely, let 1, = i, = . . . = A,. Then A is reducible. For, if not, then for some t > 0 the matrix A + tl is nonnegative and

138

JOHN Z. HEARON

irreducible and as such, according to the Frobenius Theorem ([IO], p. 65, [ll], p. 285), possesses a simple, dominant root. Thus A is permutationally similar to

where Al and A2 are square. If n = 2 we are through. Let n 3 3. Since the union of the roots of A, and A, comprises the roots of A, this same argument can be applied to whichever of A, and A, is not a scalar and the argument can be continued until we arrive at a triangular matrix, the roots of which are the diagonal entries. This completes the proof. The first corollary is a practically self-evident restatement of Theorem 1.

COROLLARY

1

Let A be real with nonnegative ofS-diagonal entries and roots I,,, &, . . ., I,,. If I, = i, = . . . = i,, then G(A) contains no cycles of order two or greater. Proof We consider the triangular form B = PAP-‘, vouched for by Theorem 1, and G(B), which is a mere relabeling of G(A). From the triangular form, for i # j, a path exists from vi to vj only if i < j. This being so, if a path exists from vi to vj then none exists from vj to vi. With what we have shown so far, the following lemma enables us to show that if A is upper Hessenberg then, provided it meets the conditions of Theorem 1, it is already in the triangular form asserted by that theorem. LEMMA

I

Let A be real with nonnegative ofl-diagonal entries. Let the shortest path between some pair of vertices of G(A) be of length n - 1, and let G(A) contain no cycles of order two or greater. Then A is permutationally similar to a lower triangular matrix with positive entries on the subdiagonal and arbitrary entries on the main diagonal. Proof If G(A) contains path v1 + v2 --f vj permuted matrix, positive and lies on only) from v1 to v,,

a path of length n - 1 then for some relabeling the + . . . + v,_~ + v, exists and in the corresponding B, the sequence bzl, bS2, bd3, . . ., bn,n_l is strictly the subdiagonal. If this path is the shortest (and hence then there is no nonzero entry below the subdiagonal,

MATRICES

139

WITH SINGLE ROOT

for if there were there would exist a path of length less than n - 1 from v1 to 0,. Further, there is no nonzero entry above the main diagonal. For, given any integers 4 and p > q + 1 etc., between 1 and n, the path u4 + 1’q+l -+ . . . + up exists (it lies on the path from v1 to v,) and if b,, # 0 we would have, contrary to hypothesis, the cycle vq -+ vqcl + . . . --f up + uq. Hence b,, = 0 for all q 3 p + 1, 1 < q, p < II. This completes the proof. We now give the result for upper Hessenberg matrices with a single root. THEOREM

2

Let A be a real upper Hessenberg entries,

matrix

with nonnegative

qfS-diagonal

positive

entries on the subdiagonal and roots ibl, ;1,, . . ., A,. If L, = R, = . . .= I,,,, then A is lower triangular.

Proqf

Since there are no zero entries on the subdiagonal, there is a path of length n - 1 in G(A) from vI to v,. Since A is upper Hessenberg, this is the shortest, in fact only, such path. By corollary 1, G(A) contains no cycles of order greater than 2. Thus by Lemma 1, the result follows and clearly the first step [relabeling of G(A)] in the proof is not necessary. This completes the proof. In terms of compartmental analysis this section can be summarized as follows: If the matrix A of an I+compartment system has a single n-fold root, then Theorem 1 tells that the compartments can be numbered in such a way that there is flow from the ith to the jth compartment only for i < j. Further if ri is the rate constant for transfer of material from the ith compartment to the environment, then for all i we must have -Izi = ri + &+iaki. For any i there can be flow to any or all of the 12 - i compartments with higher indices j > i. However, if the original matrix is required to be upper Hessenberg, then Theorem 2 asserts that if there is actual flow from the ith to the (i + 1)th compartment, for each i = 1,2, . . ,, n - 1, then if there is a single n-fold root there is, for each i, flow only from i to i + 1 and from i to the environment. Remark

We make this restriction, in Theorem 2, of a positive subdiagonal to avoid the trivialities resulting from the possibility of completely disjoint subsystems each of which is Hessenberg and can be treated separately. In the language of Ref. [5], we consider systems in which the first compartment is a source.

140

JOHN Z. HEARON

Thus, conforms

for example the most general to Theorem 1 is of the form

4-compartment

while the most general n-compartment system conforming is (cf. [12]) X, -+ X, -+ X, -+ . . . -+ x, + J 1 4. i NONNEGATIVE

NILPOTENT

system

which

to Theorem

2

MATRICES

We recall that a matrix A is nilpotent if and only if,for some q, A4 = 0. It is obvious that each root of a nilpotent matrix is zero and conversely, that any matrix with only zero roots is nilpotent (the coefficients of the characteristic equation are the elementary symmetric functions of the roots.) We also make the following observation for use in what follows: If A is nonnegative, then a given entry in any power, Aq, vanishes if and only if the corresponding entry vanishes in M%(G). In particular A is nilpotent if and only if the adjacency matrix, M,(G), of its graph is nilpotent. For, as regards the nonvanishing of any entry in any power of a nonnegative matrix, it is obviously immaterial whether the positive entries are arbitrary or are the number 1. We now show the equivalence of certain conditions on a nonnegative matrix and note that certain of these equivalences are, or by easy inference result in, known theorems in matrix or graph theory. The purpose of presenting these particular proofs is to exhibit the role which the Frobenius Theorem plays in unifying these results. THEOREM

3

Let I. II. III.

A be a square nonnegative matrix. Then the following are equivalent: G(A) contains no cycles. A is nilpotent. A is permutationally similar to a triangular matrix with zero diagonal entries. IV. A and each of its principal submatrices contain a zero row and a zero column.

Proof The proof will be to show I * II * III * I and III = IV + II. I implies II: It is obvious (and well known [7]) that if G(A) contains cycles, then M,(G) is nilpotent (we merely note that the (i,j)-entry

no of

MATRICES

WITH

SINGLE

141

ROOT

Mf;(G) is the sum of precisely as many l’s as there are sequences of length p fromj to i; if there are no cycles, every sequence is of length at most n - 1, when A is n-square). We have observed above that M,(G) is niipotent if and only if A is nilpotent. 11 implies III: If A is nilpotent, then i, = A2 = . . . = i, = 0 and Theorem I gives 111. III implies I: It is visibly true, as in the proof of Corollary 1, that the graph of the triangular form contains no cycles of order two or greater. In the present case, the diagonal entries are zero and there are no cycles of order one. III implies IV: By inspection, the triangular form, with zero main diagonal, and each of its principal submatrices contains a zero row and a zero column. IV implies II : Given lV, it is clear that the determinant of A and each principal minor is zero. The characteristic equation is thus of the form Ai2= 0. This completes the proof. In what follows denote by I and IV the negation

of I and IV respectively.

The implication r+ IV is the statement due to Marimont [13] who employed IV =z- I as the basis of a method for checking the consistency of precedence matrices. If A is an incidence of A is a permutation

matrix, then IV implies that at least one submatrix matrix. This is the theorem of Goldberg [4]. We

obtain it from IV 3 I and the observations that I implies the presence of a simple loop which in turn implies that some submatrix of M,(G) = A is a permutation matrix. We have also just given the graph theoretic statement [4: p. 201 of Goldberg’s theorem (IV implies a simple loop in G(A)) in which form the theorem, and more, is included in Lemma 5 of [14]. That II implies IV is Goldberg’s Corollary 2 [4]. The equivalence of I and III is part of Theorem 10.1 of [5], proved by graph theoretic methods. That IV follows from I includes Theorem 3.8 of [5] which is stated: I implies that in G(A) there is at least one vertex of outdegree zero (no arrows issuing from it) and at least one of indegree zero (no arrows terminating on it). Evidently IV implies at least one vertex in G(A) of outdegree zero (corresponding to a zero column) and at least one of indegree zero (corresponding to a zero row) and I implies IV.

REFERENCES 1 W. P. London and J. Z. Hearon, 2 J. Z. Hearon and W. P. London, 10

M&h. Biosci. (1972), to appear. Math. Biosci. 14, 121-134 (1972), this issue.

142

JOHN Z. HEARON

3 C. W. Sheppard, J. Theor. Biol. 33, 491 (1971). 4 K. Goldberg, J. Res. Nat. But-. Standards 63B, 19 (1959). 5 F. Harary, R. Z. Norman, and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs, Wiley, New York (1965). 6 0. Ore, Theory of Graphs, American Mathematical Society, Providence, R.I. (1962) 7 R. G. Busacker and T. L. Saaty, Finite Graphs and Networks: An Introduction with Applications, McGraw-Hill, New York (1965). 8 J. Z. Hearon, Ann. N. Y. Acad. Sci. 108,36 (1963). 9 J. Z. Hearon, Math. Piosci. 3, 31 (1968). 10 F. R. Gantmacher, Applications of the Theory of Matrices, Interscience, New York (1959). 11 P. Lancaster, Theory of Matrices, Academic, New York (1969). 12 J. L. Lebowitz and S. I. Rubinow, J. Theor. Biol. 32, 335 (1971). 13 R. B. Marimont, J. Assoc. Computing Machinery 6, 164 (1959). 14 D. Rosenblatt, Nav. Res. Log. Quart. 4, 151 (1957).