On the maximum rank of Toeplitz block matrices of blocks of a given ...

arXiv:1305.4277v1 [math.CO] 18 May 2013

On the maximum rank of Toeplitz block matrices of blocks of a given pattern Gunther Reißig∗ † 13th February 2004 Abstract We show that the maximum rank of block lower triangular Toeplitz block matrices equals their term rank if the blocks fulfill a structural condition, i.e., only the locations but not the values of their nonzeros are fixed.

1

Introduction

Associated with a sequence H of matrices, H = (Hi )i∈N∪{0} , or, equivalently, with a formal Laurent series H, ∞ X H(s) = s−i Hi , i=0

there are Toeplitz block matrices 

∗

 H0   H1 H0   Tk+1 (H) =  .. . . . .  , k ≥ 0,  . . . Hk · · · H1 H0

Otto-von-Guericke-Universit¨at Magdeburg, Chair of Systems Theory, Institut f¨ ur Automatisierungstechnik (FEIT-IFAT), PF 4120, D-39016 Magdeburg, Germany. You may find a BibTEX entry for this paper at http://www.reiszig.de/gunther/ † This is the accepted version of a paper published in Proc. 16th Intl. Symp. on Math. Th. of Networks and Systems (MTNS), Leuven, Belgium, July 5-9, 2004, B. de Moor, B. Motmans, J. Willems, P. Van Dooren, and V. Blondel, editors. ISBN 90-5682c 517-8. Katholieke Universiteit Leuven, Departement Elektrotechniek (ESAT), Heverlee, Belgium, 2004. Some errors have been corrected.

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which play a part in systems and control theory [1–5] and are also related to problems in other fields, e.g. [6, 7]. In the structural approach to the analysis of linear systems, initiated by Iri, Tsunekawa, Yajima [8] and Lin [9], one assumes that each nonzero of the matrices by which the system is described is either a fixed zero or a free parameter [10, 11], i.e., the following structural condition is imposed: Each nonzero of the matrices equals some parameter, and none of these parameters appears more than once in the matrices.

(1)

For such parameter dependent systems, one asks for generic (or structural or typical ) properties of systems, that is, properties that the system has for almost all (in one sense or another) values of the parameters [10, 11]. To see the relation to maximum ranks of Toeplitz block matrices, assume the matrices Hi depend on some parameter p ∈ Rq , Hi : Rq → Rn×m : p 7→ Hi (p) for all i ∈ N ∪ {0}, so that Tk (H) depends on p as well, Tk (H) : Rq → Rnk×mk . If H fulfills a structural condition, i.e., the parameter dependent matrix p 7→ (Hk−1 (p), . . . , H1 (p), H0 (p)) fulfills (1),

(2)

then the dependence of Tk (H) on p is analytic. This implies that the generic variants of system properties characterizable by ranks of Toeplitz block matrices and their submatrices are characterizable by the maxima over p ∈ Rq of the ranks of Tk (H)(p) and its submatrices. The difficulty is that even though H fulfills (2), Tk (H) does not fulfill (1), so that it may be very hard to determine maximum ranks if n, m or k is large. It is the purpose of this paper to show that for all k ≥ 1 the maximum rank of Tk (H) equals its term rank, i.e., that the obvious inequality    H0 (p1 )       H (p ) H (p )   1 k+1  0 2   maxq rk Tk (H)(p) ≤ max q rk   p∈R p1 ,...,pK ∈R . .. ..  ..  . .       Hk−1(pK ) · · · H1 (p2k−1 ) H0 (pk )

(3)

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is actually an equality, provided that H fulfills the structural condition (2). This result holds for matrices Hi over an arbitrary field. (Here, K = k(k − 1)/2 and rk X denotes the rank of X. The right hand side of (3) is called the term rank of Tk (H) and is denoted by rkt Tk (H).) Special cases of this result have been known for some time. The case k = 1 is due to Edmonds [12]. According to [7], the case k = 2 had been solved by Holenda and Schlegel in 1987 under the assumption that H1 is diagonal and nonsingular. Equality has been shown in [7] for general k under the assumptions that H1 is diagonal and nonsingular and that H is a pencil, i.e., H(p)(s) = H0 (p) + s−1 H1 (p). Finally, equality has been shown in [4, 13] for general k under the assumption that H is a pencil. In addition, an analogous result for one particular submatrix of Tn+1 (H) and some particular form of H has been obtained in [14], which is not a special case of the result of this paper. The proof we present in section 3 is elementary and uses only two nontrivial facts, namely, the K¨onig-Egerv´ary Theorem and a simple lemma from parametric programming. Moreover, compared to the proofs of those special cases of our result that have been obtained earlier, our proof must also be called extremely short.

2

Preliminaries

We introduce some convenient notation and collect some well-known facts. If R and C are finite sets and F is a field, we call any mapping M : R×C → F a matrix of size R × C over F. The entry in row r ∈ R and column c ∈ C of M is denoted by Mr,c . By a formal Laurent series of size R × C over F we mean a sequence H, H = (Hi )i∈N∪{0} , where the coefficients Hi of H are matrices of size R × C over F. Tk (H) denotes the block lower triangular Toeplitz block matrix associated with H, which is the matrix of size ({1, . . . , k} × R) × ({1, . . . , k} × C) over F defined by ( 0, if i < j, Tk (H)(i,r),(j,c) = (Hi−j )r,c , otherwise. Let H be a Laurent series of size R × C over F and assume R ∩ C = ∅. The bipartite graph G(H) associated with H is defined by G(H) = (R, C, E), where E = {{r, c} | r ∈ R, c ∈ C, (Hi )r,c 6= 0 for some i ∈ N ∪ {0}}. 3

Likewise, the weight function w : E → −N ∪ {0} associated with H is defined by wr,c = − min{i ∈ N ∪ {0} | (Hi )r,c 6= 0}. Let G be a bipartite graph, G = (R, C, E) with R ∩ C = ∅, with vertex set R ∪ C and edge set E ⊆ {{r, c} | r ∈ R, c ∈ C}, and let A : (R ∪ C) × E → {0, 1} be its incidence matrix [15]. Let further w : E → Z be a weight function on the edge set of G. In the following, X : E → {0, 1}, y : R → N ∪ {0}, z : C → N ∪ {0}. The bipartite cardinality matching problem B(G) is to X maximize Xr,c

(B(G) − a)

{r,c}∈E

s.t. AX ≤ 1.

Its dual, the bipartite covering problem DB(G), is to X X minimize yr + zc r∈R

(B(G) − b)

(DB(G) − a)

c∈C

  y T ≥ 1. s.t. A z

(DB(G) − b)

The bipartite cardinality-µ assignment problem A(G, w, µ) is to X maximize Xr,c wr,c (A(G, w, µ) − a) {r,c}∈E

s.t. AX ≤ 1, X Xr,c = µ.

(A(G, w, µ) − b)

(A(G, w, µ) − c)

{r,c}∈E

Its dual, DA(G, w, µ), is to minimize λµ +

X r∈R

yr +

X

zc

(DA(G, w, µ) − a)

c∈C

  y T + λ ≥ w. s.t. A z 4

(DA(G, w, µ) − b)

By the K¨onig-Egerv´ary Theorem [16], the optimal values of B(G) and DB(G) coincide, and those of A(G, w, µ) and DA(G, w, µ) coincide as well. We say that X (resp., (y, z, λ)) is admissible for problem (∗), if X (resp., (y, z, λ)) fulfills (∗−b) and, if present, (∗−c). We say that X (resp., (y, z, λ)) is optimal for problem (∗) if it is a solution of problem (∗). The following fact [17] is also useful. 2.1 Lemma. Let w be nonpositive, µ ˆ the optimal value of B(G), µ ∈ [0, µ ˆ] ∩ Z, and δ(µ) the optimal value of A(G, w, µ). Then, for given λ ∈ Z, there are y ∈ ZR and z ∈ ZC such that (y, z, λ) is a solution of DA(G, w, µ) iff the following two conditions hold: (i) µ > 0 =⇒ λ ≤ δ(µ) − δ(µ − 1), (ii) µ < µ ˆ =⇒ λ ≥ δ(µ + 1) − δ(µ).

3

Results

Unless stated otherwise, H is a Laurent series of size R×C over F. If R∩C = ∅, G denotes the bipartite graph associated with H, G = G(H) = (R, C, E), w : E → Z denotes the weight function associated with H, X ∈ {0, 1}R×C , y ∈ ZR , z ∈ ZC , λ ∈ Z, X ′ ∈ {0, 1}({1,...,k}×R)×({1,...,k}×C) , y ′ ∈ {0, 1}{1,...,k}×R , and z ′ ∈ {0, 1}{1,...,k}×C . ′ We also assume Xr,c = 0 if {r, c} ∈ / E and X(i,r),(j,c) = 0 if {(i, r), (j, c)} is not an edge of G(Tk (H)). Our method is to transform primal and dual solutions of the cardinality-µ assignment problem for G and w into primal and dual solutions, respectively, of the matching problem for G(Tk (H)). To this end, we define X ′ , y ′ and z ′ for any k ∈ N by ( 1, if Xr,c = 1 and wr,c = j − i, ′ (4a) X(i,r),(j,c) = 0, otherwise, ( 1, if i ≥ 1 − yr − λ, ′ (4b) yi,r = 0, otherwise, ( 1, if zc ≥ j, ′ (4c) zj,c = 0, otherwise, where i, j ∈ {1, . . . , k}, r ∈ R and c ∈ C.

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3.1 Proposition. Let µ ∈ N ∪ {0}, k ∈ N, and assume that X is admissible for A(G, w, µ), (y, z, λ) is admissible for DA(G, w, µ) and that (4) holds. Then X ′ is admissible for B(G(Tk (H))) and (y ′ , z ′ ) is admissible for DB(G(Tk (H))). If, in addition, the condition (µ = 0 ∨ λ ≥ −k) ∧ (µ = |R| ∨ λ = −k) holds and X and (y, z, λ) are optimal, then so are X ′ and (y ′ , z ′ ). Proof. Assume µ > 0 without loss. ′ It is obvious from (4a) that X(i,r),(j,c) ≥ 0 for all i, j ∈ {1, . . . , k}, r ∈ R and ′ c ∈ C. Further, X(i,r),(j,c) = 1 implies {r, c} ∈ E and wr,c = j − i, so that (Hi−j )r,c 6= 0 and {(i, r), (j, c)} is an edge of G(Tk (H)). ′ ′ ′ If X(i,r),(j,c) = X(i,r),(j ′ ,c′ ) = 1, then Xr,c = Xr,c′ = 1 by (4a), and hence, c = c by admissibility of X. Further, j − i = wr,c = wr,c′ = j ′ − i by (4a), which ′ implies j = j ′ . Analogously, X(i,r),(j,c) = X(i′ ′ ,r′ ),(j,c) = 1 implies r = r ′ and i = i′ , and thus, X ′ is admissible for B(G(Tk (H))). ′ ′ Likewise, yi,r ≥ 0 and zj,c ≥ 0 for all i, j ∈ {1, . . . , k}, r ∈ R and c ∈ C by ′ ′ (4b) and (4c). If {(i, r), (j, c)} is an edge of G(Tk (H)) and yi,r + zj,c = 0, then ′ ′ {r, c} is an edge of G, wr,c ≥ j − i, and yi,r = zj,c = 0. Therefore, i ≤ −yr − λ and zc ≤ j −1 from (4b) and (4c), and hence, yr +zc +λ ≤ j −i−1 ≤ wr,c −1. This is a contradiction, as (y, z, λ) is admissible for DA(G, w, µ). Thus, (y ′ , z ′ ) is admissible for DB(G(Tk (H))). Now let X and (y, z, λ) be optimal, let µ ˆ be the optimal value of B(G), and for every µ ∈ [0, µ ˆ] ∩ Z, denote by δ(µ) the optimal value of A(G, w, µ), particularly, X X X δ(µ) = wr,c Xr,c = λµ + yr + zc . r∈R

{r,c}∈E

c∈C

If ξ−i is the number of edges {r, c} ∈ E for which Xr,c = 1 and wr,c = −i, then ∞ X δ(µ) = (−i)ξ−i , (5) i=0

µ=

∞ X

ξ−i .

(6)

i=0

Observe that i > δ(µ−1)−δ(µ) implies ξ−i = 0, for if ξ−i > 0, then Xr0 ,c0 = 1 ˜ and wr0 ,c0 = −i < δ(µ) − δ(µ − 1) for some edge {r0 , c0 } ∈ E. If we define X ˜ ˜ by Xr0 ,c0 = 0 and Xr,c = Xr,c for {r, c} = 6 {r0 , c0 }, then X ˜ r,c wr,c = δ(µ) − wr0 ,c0 > δ(µ − 1), δ(µ − 1) ≥ X {r,c}∈E

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which is a contradiction. Hence, the sums in (5) and (6) actually extend over i = 0 to k, so that X

′ X(i,r),(j,c)

= δ(µ) −

k X

(−i)ξ−1 +

i=0

(i,r),(j,c)

k X

(k − i)ξ−i

i=0

= δ(µ) + kµ.

(7)

It remains to show the identity δ(µ) + kµ =

k XX r∈R i=1

′ yi,r

+

k XX

′ zj,c ,

(8)

c∈C j=1

as the optimality of both X ′ and (y ′ , z ′ ) follows from (7), (8) and the K¨onigEgerv´ary Theorem. P ′ First, our assumption λ ≥ −k implies λ + k + yr ≥ 0, from which ki=1 yi,r = Pk ′ λ + k + yr follows by (4b), and j=1 zj,c = zc follows directly from (4c). Hence, the value of the right hand side of (8) equals X X X X (λ + k + yr ) + zc = |R|(k + λ) + yr + zc r∈R

c∈C

r∈R

c∈C

= δ(µ) + |R|(k + λ) − λµ.

(9)

Finally, the value (9) equals that of the left hand side of (8), as |R| = µ or λ = −k by assumption. 3.2 Theorem. Let R and C be finite sets and k ∈ N, let H be a formal Laurent series of size R × C over a field F, and let the coefficients of H depend on some parameter p ∈ Fq . If H fulfills the structural condition (2), then the maximum over p ∈ Fq of the rank of Tk (H)(p) equals the term rank of Tk (H), maxq rk Tk (H)(p) = rkt Tk (H). p∈F

Moreover, the maximum is attained for some p ∈ {0, 1}q and equals the number of nonzeros in Tk (H)(p) for that value of p. Proof. We may assume without loss that R and C are nonempty and disjoint. Let G and w be the bipartite graph and the weight function, respectively, associated with H, G = (R, C, E) and w : E → −N ∪ {0}. Let further µ ˆ be the optimal value of B(G), i.e., the maximum cardinality of a matching in G, and for every µ ∈ [0, µ ˆ] ∩ Z, denote by δ(µ) the optimal 7

value of A(G, w, µ). As δ is concave, there exists some µ and a solution (y, z, λ) of DA(G, w, µ) with λ = −k by Lemma 2.1. Prop. 3.1 implies that for any solution X of A(G, w, µ), the matching X ′ defined by (4a) is a maximum matching in G(Tk (H)), in other words, rkt Tk (H) = rk X ′ . Now choose a special value of the parameter p by setting ( 1, if wr,c = −i and Xr,c = 1, pi,r,c = 0, otherwise for all parameter components pi,r,c associated with a nonzero (Hi )r,c . Obviously, Tk (H)(p) = X ′ for this particular p, so that rk Tk (H)(p) = rkt Tk (H).

4

Conclusions

We have shown that the maximum rank of block lower triangular Toeplitz block matrices equals their term rank if the blocks fulfill a structural condition, i.e., only the locations but not the values of their nonzeros are fixed. This result holds for matrices over an arbitrary field. The proof we presented is elementary and, compared to the proofs of those special cases that have been obtained earlier, it is also extremely short. Further results related to the structural approach to the analysis of linear state space and descriptor systems that we have obtained using the same techniques will be detailed elsewhere.

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