Reachability matrix by partitioning and related boolean results

Reachability Matrix by Partitioning and Related Boolean Results by D. S. Ministry Greece

EVANGELATOS

of Transport

and Communications,

49 Sygrou

Avenue,

117 80 Athens,

ABSTRACT: The reachability matrix of a partitioned matrix is obtained, in closed form, based on the properties of the structural matrix inverse. A graph theoretic interpretation of the partitioning is given in terms of the reachability relations on the two subgraphs into which the original graph splits, corresponding to the partitioned adjacency matrix. The reachability complement of a partitioned matrix is thus defined. A result analogous to the usual method of modt$ed matrices is also presented, called here as the “structural Householder,formula” and its special case : a formula for the reachability matrix of a sum of boolean matrices. The latter is used to prove that the structural matrix inverse (A- ‘jB does not depend on which maximal permutation matrix PA included in A is taken.

I. Introduction The reachability matrix (14) is a well known concept of graph theory originating from structural models of systems (communication systems, other large-scale systems, etc.). Its properties in determining the strongly-connected components of a graph are associated with the calculation of a permutation transforming the corresponding matrix to lower block triangular form and are known in the field of decomposition of large-scale systems (59). The reachability matrix is also met in the analysis of structural modelling (l& 14), in the structural modal matrix (15) and it has been proved (16-18) to relate to the structural matrix inverse. The problem of completing a partially filled reachability matrix, and generally the implication theory and algorithms for transitive coupling is addressed in (19, 20). The present work elaborates on the reachability matrix of a partitioned matrix obtained in a closed form, in terms of the partial matrices without having to calculate large dimensioned summations of powers. This is obtained by relating previous results on the structural matrix inverse to the boolean properties of nonnegative matrices. Interesting results are proved to follow from this relationship. II. Preliminaries 1. Reachability Matrix It is well known that to a given n x n boolean matrix As there corresponds a directed graph G = (X, E), where X = {x,, x2, . . . , x,} are the n vertices and E are the arcs of the graph represented as ordered pairs of vertices, such that m(” The Franklm lnst1tute0016--0032:8Y$3.00+0.00

409

D. S. Evungelutos ajj,

= 1

if arc (xi, xi) E E

ailB = 0,

otherwise.

A path from xi to xi is a sequence of arcs (e,, e2,. . . , ey, . . . , e,,) where the terminal endpoint of arc ey is the initial endpoint of the next arc eq+ i, for every q < p. If there is a path from x, to x, we say that x, is reachable from x, and write rrj = 1. If no such path exists we say xi is not reachable from x, and write r,i = 0. It is also agreed that rii = 1. The boolean matrix with elements r,i as above is called the reachability matrix of AB and is denoted as RAg = (rii) or simply RA (where A is any matrix with the same pattern of zeros and non-zero elements as the zeros and ones of A,). It is known that R/j = (I+AJP

=

z+AB+A;+ . .

+/I;-

(1)

where Z is the identity matrix and the mth power is defined to be the product boolean matrices A . A and this product is defined as follows :

of

. Let X,, Y, be twin x m and m xp boolean boolean matrix (zjjB) with elements : ZijB

=

f k=l

XikB

matrices.

Their product

Z, is the

‘Y~,B

where the summation is the boolean “+” or “OR” operation and the product is the boolean “.” or “AND” operation. We note that zjjB = 1 if and only if there exists at least one k such that xjkB = ykja = 1. Otherwise it is zero. Using the directed graph representation of matrices, the product can be described as : (X,* YJi, = 1, if and only if there is a k such that there exists an arc from j to k and an arc from k to i. It has been proved (18) that the reachability matrices G, n x m and K, m x n, is given by

matrix of a product

of two boolean

Rex = I+GRKGK. 2. Non-negative Matrices An m x n matrix A = (ai,) is called non-negative (noted as A > 0) if all its elements are non-negative (aii > 0). A well known result from (21) for non-negative matrices is as follows : “An n x n non-negative matrix A = (a,k) always has a non-negative eigenvalue r such that the moduli of all the eigenvalues of A do not exceed r. To this ‘maximal’ eigenvalue r, there corresponds a non-negative eigenvector Ay = ry(y 2 0, y # 0). Journal

410

of the Frankhn Pergamon

Inst~tutc Press plc

Reachability The adjoint

matrix B(2) = Blk(lZ) = (AZ-A)

Matrix

by Partitioning

‘A(A) satisfies the inequalities (3)

Let it be noted that A(2) = /I”+ .‘.ao, For non-negative

matrices

hence

A(i) + co.

X, Y and their boolean (X+ Y), = x,+ (J-Y),

counterparts

(4) X,, Y,, we have

Y,

(5)

= X,Y,.

(6)

3. Structural Matrix Inverse (1618) The structural matrix inverse (A ~ ‘)B is defined such that the element (ij) is the summation of products of boolean variables produced by forming the polynomial expansion of the minor associated with a,,. Equivalently, we can say that (A- ‘)i,s matrix of order is 1 if the complementary minor of allB includes a permutation n- 1, and is 0 otherwise. The usual inverse of A would have the same pattern of zero and non-zero elements as (A ‘)B, if unrelated numbers are taken as elements of A. This is not however, in general, the case when some element of A is equal to or a function of another element or elements, or of a common parameter, in which case more zeros might be produced in the usual inverse. For A, an n x n non-negative matrix and 2 sufficiently large, the following result is valid : ((AZ-A)-‘),

= RA.

(7)

III. Results 1. Reachability Matrix by Partitioning and the Reachability Partitioned Matrix Proposition I. Given the partitioned matrix

where A, IB isn,xn,andA,,, partitioned form is

is n2 x n2, let RA be its reachability

where RI, is n 1x n, and Rz2 is n2 x n2. Then the following Vol. 326, No. 3, pp. 409-419, Printed in Great Britain

1989

Complement

relations

matrix

of a

which in

are valid : 411

D. S. Evangelatos

and also the following

Rx = RAz~+~IBRA,,AI,B

(8)

RIZ = R,,,AizrrR22 RZI = RIVIERA,,

(10)

R,, = RA,, +R,zRz,

(11)

symmetric

relations

are valid

(9)

:

(14

RI,

=RA

R,z

=

R,,A,zeRa,,

(13)

R2’

=

RA,,A~‘BR”

(14)

I,,+AI&A,~A,IB

(15)

R22 = RA~~+Rz,R,~. Proof. Considering non-negative matrices A, with the given partitioned counterpart As, and taking i, ZL,sufficiently large, we have RA =

AZ-A,, -A21

-A,2 pZ-A22

boolean

11 -’

(16)

B’

Hence R,,

(17)

= [(~Z-A,,)p’+(ilZ-A,,)p’A,2~-‘A2,(~Z-A,,)-’l~

R,2 =

W-A,‘)-‘A&‘l.

(18)

R2, =

[~%M~~~-AII)~‘IB

(19)

R22

(5-

(20)

=

‘>B

where 5

=

~Z-[A~~+A~,(AZ-A,,)-‘AI~I.

(21)

We note that I. can be taken sufficiently large (1 2 A,) so that Eqs (4) and (3) give A(n) > 0 and (AZ- A, ,)-I 3 0. Also, from (21) it is seen that ,Mcan be taken sufficiently large (p 3 p,J such that sr-’ 3 0. Hence, taking into account (5), (6) and (7), we obtain The symmetric result (12)-( 15) is obtained from

(8))(11).

RA = g

--I

1

~hM~~--n~’

3

(~Z-A22)-‘+(pZ-A22)-1A2,g-‘A12(11Z-A22)-’

i (~~-AJ%,g?>

B

where g = through 412

~Z-[AII+A’~(~Z--A~)~‘A~II

a similar reasoning. Journal

of the Franklin Pergamon

Institute press plc

Reachability Matrix

by Partitioning

Example 1

-0

1

0

0

1

0

0

0101000 0001001 Let

A =

0000100 000100

0 A0

010001

0

001010

I RA,, =

1

1

1

1

0

1

0

1

0

1

0

1I

0011’ _o

0

and

RII =

1

1

0

1

0

1

0

1

0

1

0011’ Lo

Consequently

0

1

1101100 0101100 0011111 RA =

0001100.

1

0000100 0011111 0011111 We observe

that the calculation

Vol. 326, No. 3, pp. 409419, Printed in Great Britain

1989

of the reachability

1 matrix

of a matrix

of order 413

D. S. Evangelatos

‘G n

FIG. 1. Graph corresponding

to the partitioned A.

2n x 2n, is reduced to operations between matrices and reachability matrices of order n x n. The algorithm also described in (18) (Proposition 1) is a special case of the above Proposition 1 if the order of A 22Bis taken to be n2 = 1. We also note that to obtain R, , from A, ,B we must add to it the A ,ZBRA,,AZ ,B, and take the reachability matrix of the sum. So, A, ,B + A, 2BRA,,A2 ,B, is called the reachability complement of the partitioned matrix. It can be proved that the structural Schur complement of a matrix A is included in the reachability complement of A. 2. Graph Theoretic Interpretation We can think of the graph G,, on n vertices, associated with the n x n matrix AB, as two separate graphs, G, on n, vertices, and GZ on nz vertices (n = n, +nJ where G, is associated with the A, IB and G2 is associated with matrix AzzB, together with the arcs-joining vertices of G2 to those of G ,-corresponding to A, 2Band arcsjoining vertices of G, to those of G,-corresponding to A2 lB. The reachability relation between the vertices of G, can be considered as the reachability relation on each of G, (or GJ taking into account the reachability due to paths through AlzB, AsIB and due to the other graph G2 (or G, respectively) (see Fig. 1). Equations (8)-( 11) and (15) can be written, elementwise, (r,,),.; = 1 if and only if

(8)’

(r.4,,+a,,RA,,,4,,)i, = 1 (r , Jik

= 1 if and only if there are p and m such that (rA,,)ipapm(rAk

(9)’

= 1

(r2 Jkj = 1 if and only if there are t and o such that (10)’

(r&ato(rA, Joi = 1 (r, Jli = 1 if and only if there is a k such that Journal

414

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lnst~tute Press plc

Reachability (rA,,)ij+(r12),k(r21)kj

Also, in a way equivalent that

=

Matrix

by Partitioning

(11)’

1.

to (8)‘, we have (r22)ij = 1 if and only if there is a p, such

(rAIZ)i,+(rZl)iP(r12)pj

=

1.

(13

The graph theoretic interpretation of (8)’ is that there is a path in GZ, fromj to i, due to all the arcs of G,, iff there exists a path in the graph associated with the This graph is, in fact, graph G?, with the boolean matrix A22B+A2’BRA,,A12B. addition of a fictitious graph having adjacency matrix A 2,BRA,, A, 2B.The latter has an arc (ij) iff there exists an arc (A2 r),. a path (rA, ,)p,,, completely in G, and an arc (A I 2)mj. The graph theoretic interpretation of (9)’ is that there is a path from vertex k of G2 to vertex i in G,, i.e. (r,2)jk = 1, iff there is a path totally in G,, from p to i, an arc (A, 2)pmfrom m top and a path from k to m in G2, due to the whole of G,. An analogous interpretation is also valid for (10)‘. Now, (11)’ means that there exists a path fromj to i, in G,, due to the whole of G, iff either there is a path fromj to i totally in G, or there is a k such that a path exists from k to i and a path fromj to k, where k is in G2 and i, j in G,. Analogous is the meaning of (15)’ and the interpretation of the other symmetric relations (12)) (14). 3. Structural Householder Formulae and Reachability Matrix of a Boolean Sum The Householder formula is derived from the comparison of matrix inverses by partitioning and is used extensively in estimation theory and in inverting large matrices. It is used also in Kron’s tearing (22) (F+GHK)-’ Here comparing

= Fp’-Fp’G(Hp’+KFp’G)-‘KF-‘.

(22)

Eqs (8) and (15) and taking into account

(13) and (14), we obtain

R ALIB+AZISRA,,AIZB = RA,,+RA~,A~IBRA,,,~A,~~R~~~A~,~A~IBRA~~ which in terms of the structural

(23)

inverse is written :

([I+A~~B+A~IB[(I+AIIB)-‘~BAI~BI-’}B =

[(~+A~~~)-‘IB+[(~+A~~~)~‘IBA~IB

X{[~+AI’B+A’~B[(~+A~~B)-‘~BA~IB~-’}BA’~~[(~+A~~B)~‘~B. We call (23) and (24) the structural we obtain R Az,,+*z~nA,,, =

Householder

(24)

formula

RA,,+RA,,A~‘BRA,~~R,!~A,,A

I. From (23) for A l I = 1,

‘zBRA,,

hence R A+B’C= RA

+RAB’&R~B~CRA

(25)

and for B’ = BK Vol. 326, No. 3, pp. 409419, ‘989 Printed in Great Britain

415

D. S. Evangelatos R A+BKC

=

RA+RABK&x~BKCRA

(26)

which also can be written in terms of structural inverses. We may call (26) the structural Householder formula II. Note that from (23) or (26), Eq. (2) follows as a special case ; from (23) for A,, = Z, AZ2 = Z and from (26) for A = Z and K = I. Putting C = Zand B’ = B, in (25) gives R A+B=R~+R~BRR,~R~

&BRA.

=

(27)

Also, from (25) for B = Z, we have R A+C From the commutative conclude that

property

=

RARCR,.

(28)

of the A and B matrices

R A+B

-

RB+A

=

in the boolean

sum, we

(29)

&,ARB

and R A+C- -R

C+A

(30)

=&RAP+

i.e. the following result is obtained : Proposition 2. Given two boolean matrices A, B, each n x n, the reachability matrix of the summation of A and B is given as the product of the reachability matrices of RAB and A. Also, R AtB

=

&,ARB

=

RARBR,

=

RBRAR,

=

(31)

&BRA.

Note that (23) can be derived from (26) in view of (31). Consequently (26) are one formula, the boolean Householder formula. Example 2 Let

A

z.rz

I

001 001

(23) and

I.

00 1

Here RA = A

RAB

&BRA

1

0

0

0

0

1

1

1 =

:0011 i

0

0

11

RA+B.

1 Journal

416

of the

Franklm Pergamon

Institute

Press plc

Reachability

,

Matrix

by Partitioning

Similarly

RB =

hence

I 1 1

&,A&

=

0

0

0

0

1

1

1

0

0

1

1

0

0

11

=RA+B.

4. Uniqueness of the Structural Matrix Inverse On the basis of Proposition 2 we can have a new proof of the Corollary 2 of (17), i.e. that for every P,, P2 permutation matrices of full rank included in the boolean matrix A, we have P:RApr

= P;RAPp

(32)

Indeed, since P, & A, P2 & A, we have that A can be put in the form of a boolean sum, as follows : A = P,+Pz+K. Hence PTRAP: = PTR~p,+pzfKjP:. = P:RP~P:+PTRP,P:RK(P:R,*,~K(PTRP,P~) 4 fi(PTRP2P:> K). From P,T = (P- ‘)B, we have from (17) RP = R; for any permutation

matrix

P, and also P+R,

= PR,

= R, = R;.

Hence

Vol. 326, No. 3, pp. 409-419, Printed in Great Britain

1989

417

D. S. Evangelatos i.e. P:RPZP: = P;Rplp;. Consequently f,

QED

(P?b2~:,K) = .fi(f’;&,.:, K) = P:R,,>

ZV. Discussion and Conclusions It has been seen that the reachability matrix of a partitioned matrix can be obtained in a closed and partitioned form, in terms of products of the individual lower up to half order reachability matrices and the partial matrices themselves. This is a considerable improvement if, for example, we think of the composite interconnection matrix of a system (23), since we can generally obtain the interactions between states, inputs and outputs for any system form, and not only for the special form already studied having no feedback. The results also provide a characterization of the reachability matrix of the boolean summation of two n x n matrices and give rise to the structural inverse of the summation of a matrix and a product, which corresponds to the problem of calculating the reachability matrix of linearly parameterized systems (24). It also generalizes the Householder formula into the domain of boolean matrices. A previous corollary based on the uniqueness of the structural matrix inverse is also proved using the properties of the reachability matrix of permutations and of the matrix sum. Acknowledgement The author gratefully acknowledges discussions with Professor H. Nicholson, for his encouragement, advice and suggestions in the course of this research.

References (1) N. Christofides,

“Graph Theory : An Algorithmic Approach”, Academic Press, London, 1975. (2) F. Harary, “A graph theoretic method for the complete reduction of a matrix with a view toward finding its eigenvalues”, J. Math. P&s., Vol. 38, p. 104, 1959. (3) F. Harary, “A graph theoretic approach to matrix inversion by partitioning”, Num. Math., Vol. 4, p. 128, 1962. (4) F. Harary, R. Z. Norman and D. Cartwright, “Structural Models : An Introduction to the Theory of Directed Graphs”, Wiley, New York, 1965. (5) F. J. Evans, C. Schizas and J. Chan, “Control system design using graphical decomposition techniques”, Proc. ZEE, Vol. 128, Pt. D, p. 77, 1981. (6) 0. 1. Franksen, P. Falster and F. J. Evans, “Qualitative Aspects of Large Scale Systems-Developing Design Rules Using APL,” Springer, Berlin, 1979. (7) A. K. Kevorkian and J. Snoek, “Decomposition in Large Scale Systems, in “Decomposition of Large-Scale Problems” (Ed. D. M. Himmelblau), North Holland, New York. 1973.

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Reachability

Matrix

by Partitioning

(8) A. Titli (Ed.), “Analyse et Commande des Systemes Complexes”, Cepadues, Toulouse, 1979. (9) M. Vidyasagar, “Input-Output Analysis of Large Scale Interconnected Systems”, Springer, Berlin, 1981. (10) J. N. Warfield, “On arranging elements of a hierarchy in graphic form”, IEEE Trans. Syst. Man. Cyber., Vol. 3, p. 121, 1973. (11) J. N. Warfield, “Binary matrices in system modelling”, IEEE Trans. Syst. Man. Cyber., Vol. 3, p. 441, 1973. (12) J. N. Warfield, “Developing subsystem matrices in structural modelling”, IEEE Trans. Syst. Man. Cyber., Vol. 4, p. 74, 1974. (13) J. N. Warfield, “Developing interconnection matrices in structural modelling”, IEEE Trans. Syst. Man. Cyber., Vol. 4, p. 81, 1974. (14) J. N. Warfield, “Towards interpretation of complex structural models”, IEEE Trans. Syst. Man. Cyber., Vol. 4, p. 405, 1974. (15) J. P. Norton, “Structural zeros in the modal matrix and its inverse”, IEEE Trans. Aut. Control, Vol. 25, p. 980, 1980. (16) D. S. Evangelatos, “Analysis of Structure and Decomposition of Large Scale Systems”, Ph.D. Thesis, University of Sheffield, U.K., Dec. 1984. (17) D. S. Evangelatos and H. Nicholson, “Graph theoretic structure of the matrix inverse relating to large scale systems”, Int. J. Control, Vol. 41, p. 499, 1985. matrix and relation to large scale (18) D. S. Evangelatos and H. Nicholson, “Reachability systems”, Znt. J. Control, Vol. 47, p. 1163, 1988. (19) A. Ouchi, M. Kurihara and J. Kaji, “A theorem and a procedure for the complete implication matrix of system interconnection matrices”, IEEE Trans. Syst. Man. Cyber., Vol. 14, p. 545, 1984. (20) M. Venkatesan, “Development and storage of interpretive structural models”, IEEE Trans. Syst. Man. Cyber., Vol. 14, p. 550, 1984. (21) F. R. Gantmacher, “The Theory of Matrices”, Chelsea, New York, 1959. (22) H. Nicholson, “Structure of Interconnected Systems”, Peter Peregrinus, Stevenage, U.K., 1978. (23) D. D. Siljak, “Large Scale Dynamic Systems”, North Holland, Amsterdam, 1978. controllable and structurally canonical (24) J. P. Corfmat and A. S. Morse, “Structurally systems”, IEEE Trans. Aut. Control, Vol. 21, p. 129, 1976.

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