ON THE BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES IAIN S. DUFF and BORA UÇAR Technical Repot: No: TR/PA/08/26 CERFACS 42 av. Gaspad Coiolis, 31057 Toulouse, Cedex 1, Fance. Available at http://www.cefacs.f/algo/epots/ Date: Apil 2, 2008.
ON THE BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES IAIN S. DUFF AND BORA UÇAR Abstact. We pesent some obsevations on the block tiangula fom (btf) of symmetic, stuctually ank deficient, squae, spase matices. As the matix is squae and stuctually ank deficient, its canonical btf has at least one undedetemined and one ovedetemined block. We pove that these blocks ae tansposes of each othe. We futhe pove that the squae block of the canonical btf, if pesent, has a special fine stuctue. These findings help us ecove symmety aound the anti-diagonal in the block tiangula matix. We also visit the full ank symmetic case. Key wods. spase matices, block tiangula fom, Dulmage-Mendelsohn decomposition, maximum cadinality matchings AMS subject classifications. 05C50, 05C70, 05D15, 65F50 1. Intoduction. We ae inteested in the block tiangula fom (btf) of stuctually ank deficient, symmetic, spase matices. Unless othewise stated, A is always such a matix with no all-zeo ows o columns. The block tiangula fom is based on a canonical decomposition of bipatite gaphs known as the Dulmage- Mendelsohn decomposition (see [9] fo a detailed account). When pemuted into the block tiangula fom, the matix A assumes the fom H C S C V C H R A H S R O A S. (1.1) V R O O A V Hee, H R, S R, and V R ae sets of ows, and H C, S C, and V C ae sets of columns. As we shall see, the thee diagonal blocks ae of special impotance. The block A H, fomed by the ows in the set H R and the columns in the set H C, is undedetemined; the block A S, fomed by the ows in the set S R and the columns in the set S C, is squae; the block A V, fomed by the ows in the set V R and the columns in the set V C, is ovedetemined. As in [9], we will call these thee blocks hoizontal, squae, and vetical, espectively. In (1.1), thee ae no nonzeo enties in the sub-diagonal blocks shown as O. In the following two subsections, we povide the eade with definitions (mostly standad), and necessay backgound mateial fom Duff, Eisman, and Reid [3, Chapte 6], Pothen and Fan [9], and Pothen [8, Section 2.7] on the computation and popeties of the btf. The computation of the btf is based on maximum cadinality matchings, o just maximum matchings, in bipatite gaphs (these ae discussed in Sections 1.1 and 1.2). We discuss two tansfomations on maximum matchings of symmetic matices in Section 2. One of the tansfomations is based on [4]; the othe is based on the notion of cycles of a pemutation, and to the best of ou knowledge is discussed and used fo the fist time in this pape. We use these tansfomations to show that fo a symmetic matix thee is a maximum matching with some special Atlas Cente, RAL, Oxon, OX11 0QX, England (i.s.duff@l.ac.uk). CERFACS, 42 Av. G. Coiolis, 31057, Toulouse, Fance (duff@cefacs.f, uboa@cefacs.f). This wok was suppoted by Agence Nationale de la Recheche, though SOLSTICE poject ANR-06-CIS6-010. 1
2 DUFF AND UÇAR popeties. In Section 3, we fomally state ou main theoem on the btf of symmetic matices. The theoem establishes equivalence elations between the pai of sets H R and V C, the pai of sets S R and S C, and the pai of sets V R and H C. In the same section, we pove ou main theoem by exploiting the popeties discussed in Section 2. 1.1. Definitions. As is common, we associate a bipatite gaph G = (R C, E) with the n n matix A, whee R = { 1,..., n } and C = {c 1,..., c n } ae the two sets of the vetex bipatition, and E is the set of edges. Hee, the vetices in R and C coespond to the ows and the columns of A, espectively, such that ( i, c j ) E if and only if a ij = 1. Fo a given i {1,..., n}, the ow i and the column c i ae efeed to as symmetic countepats of each othe. Similaly, the edges ( i, c j ) and ( j, c i ) ae called symmetic countepats of each othe. When necessay, we will make it clea whethe a vetex is a ow o a column vetex. An edge ( i, c j ) E is said to be incident on the vetices i and c j. Two vetices ae called adjacent if thee is an edge incident on both. The set of vetices that ae adjacent to a vetex v ae called its neighbous and ae indicated by adj(v). A path is a sequence of edges of the fom ( (v 0, v 1 ), (v 1, v 2 ),..., (v k 1, v k ) ) (. A cycle is a sequence of edges of the fom (v0, v 1 ), (v 1, v 2 ),..., (v k 1, v k ) ) whee v k = v 0. A set of edges M is a matching if no two edges in M ae incident on the same vetex. In matix tems, a matching coesponds to a set of nonzeo enties no two in a common ow o column. A vetex is said to be matched (with espect to a given matching) if thee is an edge in the matching incident on the vetex, and to be unmatched othewise. Given a matching M, an M-altenating path is a path whose edges ae altenately in M and not in M. We use the notation u M v to denote that vetex u eaches vetex v with an M-altenating path. Note that this is a bidiectional elation in an undiected gaph: if u M v, then v M u. An altenating path is called an augmenting path, if it stats and ends at unmatched vetices. The cadinality of a matching is the numbe of edges in it. A maximum cadinality matching o a maximum matching is a matching of maximum cadinality. Given a bipatite gaph G and a matching M, a necessay and sufficient condition fo M to be of maximum cadinality is that thee is no M-augmenting path in G (the esult is due Bege and summaized in diffeent places, see fo example [6, Chapte 1]). We use mate(v), to denote the vetex matched to the vetex v in a matching M, e.g., if mate( i ) = c j, then we also have mate(c j ) = i. We use, to diffeentiate a matching edge fom an odinay edge, e.g., we use i, c j o c j, i to denote that the ow i is matched to the column c j. We say a vetex set X is completely matched to anothe one Y, if fo all x X, we have mate(x) Y ; fo claity we note that X Y, whee denotes the cadinality of a set. Some of the definitions in this paagaph can be found in [7]. Let A be an n n matix, I and J be two subsets of {1,..., n}. The matix fomed by selecting the ows and columns indexed by I and J, espectively, is called a submatix of A confined to the ows in I and the columns in J. The matix A is said to be patly decomposable if it contains an s (n s) zeo submatix. Moe explicitly, A is patly decomposable if thee exist pemutation matices P and Q such that ( ) B C P AQ =, O D with B and D being squae. If A contains no s (n s) zeo submatix fo s = 1,..., n 1, then it is called fully indecomposable, also called ieducible [3]. We
BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES 3 note fo late use that an n n symmetic matix A, whee n > 2 and n is odd, a ij = a ji 0 fo i = 1,..., n and j i + 1 (mod n), and a ij = a ji = 0 elsewhee, is ieducible. The bipatite gaph of this matix is a cycle on 2n vetices with n ow vetices and n column vetices. Any n n matix B whose spasity stuctue is a supeset of that of A, i.e., b ij 0 if a ij 0, is also fully indecomposable. Ou last definition is fo diected gaphs. A vetex v is said to be eachable fom anothe vetex u if thee is a diected path fom u to v. A stongly connected component of a diected gaph G = (V, E) is a maximal set of vetices U V such that evey pai of vetices in U ae eachable fom each othe. 1.2. Computation and popeties of the block tiangula fom. Given a maximum matching M, the btf of a matix (of any shape o symmety) can be computed using the following equations (the equations ae ephased fom [8, Section 2.7]): U C = {c C : c is unmatched} U R = { R : is unmatched} H R = { R : M u fo some u U C } V C = {c C : c M u fo some u U R } H C = {c C : c M u fo some u U C } V R = { R : M u fo some u U R } H C = U C H C V R = U R V R S R = R \ (H R V R ) S C = C \ (H C V C ). The algoithms to compute the btf of a matix wee discussed some time ago, see fo example [5, 9]. We pesent an essential pat of those algoithms in a diffeent fom below. Algoithm 1 shows how to find the set of ows H R and the set of columns H C of the hoizontal block. The algoithm gows the ow set H R and the column set H C by unning a gaph seach algoithm. At a column vetex c known to be in H C (whose adjacency is not exploed yet), it adds all neighbouing ows to H R. At a ow vetex known to be in H R, the algoithm only visits the column v = mate() and adds v to H C if it is not aleady thee. Algoithm 1 Algoithm to find hoizontal block of the btf Input M: a maximum matching; mate(v) giving the mate of a vetex v Output H R and H C: the set of ows and columns of the hoizontal block 1: H R H C 2: U {unmatched columns} 3: while U do 4: Pick a column vetex c U and set U U \ {c} 5: H C H C {c} 6: fo each ow vetex adj(c) \ H R do 7: H R H R {} 8: v mate() should exist; othewise flags an augmenting path 9: if column v / U H C then 10: U U {v} A simila algoithm is un to find the ows and columns in the vetical block. In this case, at a column vetex in V C, only its mate is visited and added to V R, if necessay; at a ow vetex in V R, the neighbouing columns ae added to V C, wheneve necessay. Afte finding the ows and columns of the hoizontal and vetical blocks, the emaining ows and columns ae maked to be in the sets S R and S C, espectively.
4 DUFF AND UÇAR We note the following popeties of the block tiangula fom without poving them. The poofs can be found in [2], [3, Chapte 6] and [8, Section 2.7]. These popeties hold fo any matix. Fact 1.1. The ows in H R ae completely matched to the columns in H C. The columns in S C ae completely matched to the ows in S R and vice vesa. The columns in V C ae completely matched to the ows in V R. Fact 1.2. The block tiangula fom is unique. In othe wods, any given maximum matching yields the same sets H R, H C, S R, S C, V R, and V C. The pevious two popeties also imply that all enties of a maximum matching should eside in the diagonal blocks of the btf. Fact 1.3. In the block tiangula fom of a stuctually ank deficient, squae matix (not necessaily symmetic), the hoizontal and vetical blocks both should be pesent. The squae block may be missing. It may be possible to decompose the thee diagonal blocks A H, A S, and A V futhe into smalle submatices, esulting in a fine decomposition. In the fine decomposition, the hoizontal and vetical blocks have block diagonal stuctue whee the individual diagonal blocks ae hoizontal and vetical, espectively. The fine decomposition of the squae block A S is obtained by identifying ieducible blocks. Pothen and Fan [9] list the popeties of and give an algoithm to compute the fine decomposition of the squae block. Hee, we summaize some of the popeties that we will need in the est of the pape. Let p be the numbe of ieducible blocks (all of them squae) in the fine decomposition of A S, and let R i and C i be the set of ows and the set of columns in the ith block fo i = 1,..., p. The ows in R i ae matched to the columns in C i. The sets R i and C i fo i = 1,..., p ae unique they ae independent of the choice of maximum matching [2]. The blocks cannot be combined to yield anothe decomposition satisfying the popeties given above. 2. Two tansfomations. As discussed in the pevious section, the btf is unique and can be obtained using any maximum matching. In this section, we discuss two pocesses that tansfom any maximum matching in the bipatite gaph of a symmetic matix into anothe one that has some special popeties. Given a matching in the bipatite gaph of a squae matix, we define an m-path as a sequence of edges of the fom ( i, c j, j,,...,, c k, k, c l ), whee each edge is in the matching. Afte a matching edge k, c l, the next edge to be included is eithe of the fom l, o, c k, if any of them exists. Note that an m-path is not necessaily a path in the bipatite gaph, as the two consecutive matching edges, c k and k, ae not necessaily connected by the edge (c k, k ). An m-path stats at a ow vetex and ends at a column vetex. If l = i in the above example, then we have an m-cycle. An m- path is called open if the symmetic countepats of the stat and end vetices of the path ae unmatched. An open m-path can be identified with the following pocess: stat fom a matched ow i whee the column c i is unmatched, visit the column c j = mate( i ), and continue fom ow j if it is matched. Note that any matching in a squae matix can be decomposed into m-cycles and open m-paths. 2.1. Automophic maximum matchings. We define a matching to be automophic if it matches a set of ows to the coesponding set of columns. That is, fo a matching M to be automophic, wheneve i, c j M, the column c i and the ow j should be matched by M. We estate the following lemma fom [4]. Lemma 2.1 (Popety 4.2 of [4]). Let A be a symmetic matix and M be a maximum matching. Let I and J be the set of ows and columns matched by M, i.e.,
BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES 5 I = { i : i, M} and J = {c j :, c j M}. Then, thee is an automophic maximum matching M that matches the set of ows I to the set of columns that ae symmetic countepats of the ows in I. Equivalently, thee is an automophic maximum matching that matches the set of columns J to the set of ows that ae symmetic countepats of the columns in J. We summaize the main points of the poof of the lemma fo completeness. Duff and Palet fist note that edges of a maximum matching ae eithe in m-cycles o in open m-paths. Since the m-cycles ae aleady automophic, they investigate the open m-paths. They note that if an open m-path is fomed using the ow vetices in the set I and the column vetices in the set J, then I\J = J\I = 1. They show that using the edges symmetic to those of the open m-path (the matix is symmetic), it is possible to completely match the set of ows in I to the set of columns in (I \ J) (I J) = I. Notice that since the m-cycles of the oiginal maximum matching ae kept intact, and fo each open m-path an automophic matching with the same cadinality is obtained, they end up with an automophic maximum matching. 2.2. Pemutation cycles of an automophic matching. Let M be an automophic matching. Since M matches a set of ows to the set of coesponding columns, its edges eside in m-cycles, i.e., thee is no open m-path. An automophic matching fom I to I can be peceived as a pemutation of the set I in an algebaic sense (a one-to-one and onto function). By stating fom an element of the set I and by applying the pemutation until the stating element is seen again, we can obtain cycles of the pemutation (fo moe on cycles of a pemutation see [1, Section ( 1.5]). Similaly, by following the matching edges of an automophic matching M as i, c j, j,,...,, c i ), we can obtain the cycles of M. Due to this coespondence, we efe to the cycles of an automophic matching as the pemutation cycles. Note that pemutation cycles ae also m-cycles and theefoe they do not necessaily coespond to odinay cycles in the undelying bipatite gaph. Conside fo example the pemutation cycle ( i, c j, j, c i ). If a ii o a jj is zeo in A, then we do not have a cycle in the gaph; we only have a pemutation cycle. The length of a pemutation cycle is the numbe of matching edges in it. The length 2 pemutation cycles, also called tanspositions [7, p.11], ae of the fom ( i, c j, j, c i ) and ae of special impotance. Figue 2.1 displays pemutation cycles of length 1 to 4 in a hypothetical example. An odd pemutation cycle is of odd length, and an even pemutation cycle is of even length. Note that the edges of an odd pemutation cycle (with length geate than one) when put togethe with thei symmetic countepats fom a unique odinay cycle in the bipatite gaph of A. Note also that any pai of a ow and a column in an odd pemutation cycle ae eachable fom each othe via two altenating paths: one stating and ending with a matching edge, the othe stating and ending with an odinay edge. Fo an even pemutation cycle of length k, adding the symmetic edges patitions the pemutation cycle into two odinay cycles each having k/2 matching edges and k/2 non-matching edges, whee the ow vetices in one cycle ae symmetic countepats of the column vetices in the othe one; the two cycles may be connected in the bipatite gaph due to existence of othe edges, but we ae not inteested in this possibility. Conside, fo example, the length 4 pemutation cycle ( i, c j, j, c k, k, c l, l, c i ) shown in Fig. 2.1. The pemutation ( cycle is split between two odinay cycles in the bipatite gaph of A: i, c j,(c j, k ), k, c l,(c l, i ) ) and ( j, c k,(c k, l ), l, c i,(c i, j ) ). As seen, each of these cycles contain 2 matching edges and 2 non-matching edges, and the ow vetices in one cycle ae symmetic countepats of the column vetices in the othe.
6 DUFF AND UÇAR i c j i c j j c k i c j j c k k c l i c i j c i k c i l c i Fig. 2.1. Pemutation cycles of an automophic matching. The matching edges ae shown with bold solid lines. The othe edges, shown with dashed lines, ae pesent because of the symmety of the matix. Matching edges of the fom i, c i give a pemutation cycle of length 1 (fist subfigue); two matching edges of the fom ` i, c j, j, c i give a pemutation cycle of length 2 (second subfigue); length 3 (thid subfigue) and length 4 (fouth subfigue) pemutation cycles ae also shown. Fom an automophic matching M, we constuct anothe automophic matching M which is composed of odd length pemutation cycles and length 2 pemutation cycles. We poceed as follows. Fist, all edges of M that fom an odd pemutation cycle ae copied into M, e.g., fo a length 3 pemutation cycle of the fom ( i, c j, j, c k, k, c i ), these thee edges ae copied into M. Then, even length pemutation cycles of M ae decomposed into length 2 pemutation cycles, and these length 2 pemutation cycles ae added to M such that if i, c j M, then j, c i M. As noted above, the even length pemutation cycles ae split between two odinay cycles in the bipatite gaph when the symmetic edges ae consideed. By altenating the status of the edges accoding to the matching in one of the cycles, we can obtain length 2 pemutation cycles. The decomposition of an even pemutation cycle into length 2 pemutation cycles is best seen in matix tems. Conside the matching shown on the left below 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4. (2.1) 5 6 In the matix on the left of (2.1), the oiginal matching, with matching enties maked by, is automophic and coesponds to a length 6 pemutation cycle. Using the enties symmetic to the matching enties, shown with, we obtain two odinay cycles: fist ( 1, c 2,(c 2, 3 ), 3, c 4,(c 4, 5 ), 5, c 6, (c 6, 1 ) ) and second ( 2, c 3, (c 3, 4 ), 4, c 5, (c 5, 6 ), 6, c 1, (c 1, 2 ) ) which shae the oiginal matching edges evenly. Now by taking the second cycle and altenating the status of the edges accoding to the matching, we obtain ( ( 2, c 3 ), c 3, 4, ( 4, c 5 ), c 5, 6, ( 6, c 1 ), c 1, 2 ). As seen, the new set of matching edges fom thee length 2 pemutation cycles: ( 1, c 2, 2, c 1 ), ( 3, c 4, 4, c 3 ), and ( 5, c 6, 6, c 5 ). The esulting matching is shown on the ight of (2.1). As is clea, this tansfomation does not change the cadinality of the automophic matching and hence M is of maximum cadinality.
BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES 7 c i 3. The block tiangula fom of symmetic matices. As shown in the pevious section, any maximum matching in the bipatite gaph of a stuctually singula, symmetic matix A can be tansfomed into an automophic one consisting of only odd pemutation cycles and length 2 pemutation cycles. Theefoe, we assume that we have a matching M with those popeties. We ecall Fact 1.2 the uniqueness of the sets H R, S R, V R, H C, S C, and V C of the block tiangula fom of a matix. Befoe we pove ou main theoem, we stat with a seies of lemmata. Lemma 3.1. The odd pemutation cycles ae confined to only one diagonal block of the block tiangula fom. Poof. Pemutation cycles of length 1 ae tivially confined to only one block. Recall fom Section 2.2 that an odd pemutation cycle is a pat of an odinay cycle. Since the matching edges within the odinay cycle ae in the diagonal blocks of the btf, having those nonzeos staddle moe than one diagonal block can only be possible if the submatix confined to the ows and columns of the cycle is patly decomposable. Howeve, as noted towads the end of Section 1.1, matices whose spasity stuctue coesponds to a supeset of odd length cycles ae fully indecomposable. We have a esult which is stonge than the pevious lemma. Lemma 3.2. The odd pemutation cycles ae confined to the squae block. Poof. Fom Lemma 3.1, an odd length pemutation cycle is confined to a single block. Take an odd length pemutation cycle C = ( i, c j, j,,...,, c i ) and suppose, fo the sake of contadiction, that C is in the hoizontal block. Since all ow vetices in C ae in H R, each one of these vetices eaches an unmatched column with an altenating path. Suppose ow i eaches, with an altenating path, an unmatched M column c u, i.e., i cu without going though othe vetices in the pemutation cycle ( C (the latte assumption is not weake but gives a cleane agument). Let P = (i, c l ), c l, i+1,...,, i+t, ( i+t, c u ) ) be that altenating path. Due to the symmety of the matix, the same path exist in the evese diection fom ow u to column c i. That is we have the path P T = ( ( u, c i+t ), (c i+t, ),..., (c i+1, l ), ( l, c i ) ). We now show that P T is an altenating path, i.e., P T = ( ( u, c i+t ), c i+t,,..., c i+1, l,( l, c i ) ). Note that since u and c u ae both unmatched, P T being an altenating path implies that the path ( M M M ( u, c i+t ), c i+t ci i i+t, ( i+t, c u ) ) is an augmenting path, contadicting the assumption that M is a maximum matching. We fist note that M i as c i and i ae in an odd pemutation cycle. Note that since M is automophic, ow u is not matched. Conside the last ow vetex i+t in P. Column c i+t is the fist column vetex in P T. Since u is not matched, c i+t should have a mate (othewise M would not be a maximum matching). Theefoe, c i+t is in the vetical block (being a matched vetex eaching an unmatched ow). Now, since i+t is in the hoizontal block and c i+t is in the vetical one, due to Lemma 3.1 the vetices i+t and c i+t cannot be in an odd pemutation cycle. Theefoe, they ae in a length 2 pemutation cycle. That is, if mate( i+t ) = c x, then mate(c i+t ) = x. Conside the next column vetex c i+t 1 in P T. It should have a mate, othewise an augmenting M path u ci+t 1 exists, and should be in the vetical block. With the same easoning as above, it is matched to the ow that coesponds to the mate of ow i+t 1 in P. Theefoe, the path P T is an altenating path symmetic to P. Figue 3.1 displays the aguments fo a length 3 pemutation cycle C = ( i, c j, j, c k, k, c i ) and an altenating path P = ( ( i, c l ), c l, m, ( m, c u ) ). With simila aguments, it can be shown that odd pemutation cycles cannot be confined to the vetical block. Theefoe, the odd pemutation cycles ae confined to
8 DUFF AND UÇAR c u m c l i c j j c k k c i l c m u Fig. 3.1. An example fo the poof of Lemma 3.2. The solid bold lines coespond to edges of a maximum matching. A length 3 pemutation cycle C = ` i, c j, j, c k, k, c i is shown; the solid lines epesent the edges symmetic to that of the pemutation cycle. Column c u is not matched and eachable fom ow i with the altenating path P = `( i, c l ), c l, m, ( m, c u). The symmetic path P T is shown with dashed lines. Row u is not matched, as the matching is automophic. Column c m should have a mate, and since it eaches an unmatched ow, it should be in the vetical block. It is shown in the poof that the column c m should have been matched to l ; this matching edge is shown with a dashed bold line. Theefoe, the path `(u, c m), c m, l, ( l, c i ), c i, k, ( k, c j ), c j, i, ( i, c l ), c l, m, ( m, c u) is an augmenting path, contadicting the fact that M is a maximum matching. the squae block. Coollay 3.3. Fo each i, c j M in the hoizontal block, we have j, c i M. Similaly, fo each k, c l M in the vetical block, we have l, c k M. We have a efinement of the pevious coollay. Lemma 3.4. The length 2 pemutation cycles ae not contained entiely in the hoizontal o vetical blocks. Poof. We pove the lemma fo the hoizontal block, that is we show that length 2 pemutation cycles ae not contained in the hoizontal block; the vetical block case is simila. Suppose, fo the sake of contadiction, i, c j M and its symmetic countepat j, c i M ae in the hoizontal block. As in the poof of Lemma 3.2, we take an unmatched column c u that is eachable fom ow i with an altenating path. Again, due to M being automophic, ow u is not matched. Howeve, as in the poof of Lemma 3.2, we have an altenating path fom column c i to unmatched ow u, contadicting the fact that c i is in the hoizontal block. We ae now eady to state and pove the following theoem egading the block tiangula fom of a stuctually ank deficient symmetic matix. Theoem 3.5. Given a stuctually ank deficient symmetic matix A, let H R, S R, V R, H C, S C, V C be the sets in the block tiangula fom of A. Then, the set of hoizontal ows H R is equal to the set of vetical columns V C ; the set of squae ows S R is equal to the squae columns S C ; the set of vetical ows V R is equal to the set of hoizontal columns H C. Poof. Since A is squae and ank deficient, we know that both hoizontal and vetical blocks ae pesent in the block tiangula fom. The squae block may be
BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES 9 K J SC I I SR J L Fig. 3.2. A matching is shown by the slanted line; the ows in I ae matched to columns in J, and the columns in I ae matched to ows in J. The bodes of the hoizontal, squae, and vetical blocks ae shown with solid lines. The dashed lines divide the set of columns in the hoizontal block and the set of ows in the vetical block into two sets. The sets K and L ae equal as the set of unmatched columns is equal to the set of unmatched ows. missing. Conside a matching edge i, c j in the hoizontal block. As shown in Lemma 3.2, it is not in an odd pemutation cycle and hence, as noted in Coollay 3.3, j, c i M. We know fom Lemma 3.4 that j, c i M is not in the hoizontal block. Two cases emain to be investigated: j, c i is eithe in the squae block o in the vetical block. Fo the sake of contadiction, suppose j, c i is in the squae block. As in the poof of Lemma 3.2, we take an unmatched column c u that is eachable fom ow i with an altenating path. Again, due to M being automophic, ow u is not matched. Howeve, as in the poof of Lemma 3.2, we have an altenating path fom column c i to the unmatched ow u, contadicting the fact that c i is in the squae block. Simila aguments can be used to show that fo a k, c l in the vetical block, l, c k is in the hoizontal block. Theefoe, a matching edge i, c j is in the hoizontal block if and only if the matching edge j, c i is in the vetical block. We have established two esults. Fist, the set H R is equal to the set V C. Second, the set of columns that ae matched to the ows in H R is equal to the set of ows that ae matched to the columns in V C. These equivalence elations ae shown in Fig. 3.2. As the matching is automophic, the set of unmatched columns (K in the figue) is equal to the set of unmatched ows (L in the figue); we have thus established the equivalence between the sets H C and V R. Since the matix is squae, the set of emaining ows S R is equal to the set of emaining columns S C. Once the equivalences between the ow and column sets ae established, it is easy to ecove a stuctual symmety in the block tiangula fom. Coollay 3.6. The block tiangula fom of a stuctually singula, symmetic matix can be pemuted to be symmetic aound the anti-diagonal. This can be achieved by fixing a pemutation of the ows in H R and S R, and the columns in H C, and then by eoganizing V R, S C and V C such that the evese ode within these late blocks match those of H C, S R, and H R, espectively. It is possible to efine this fom by looking at the fine stuctue of the squae block A S. We fist need the following lemma. Lemma 3.7. Let R i and C i be the set of ows and the set of columns of the ith ieducible block in the fine decomposition of the squae block A S. Then, eithe R i = C i ; o R i C i = and thee exist an ieducible block j in the fine decomposition
10 DUFF AND UÇAR of A S with R j = C i and C j = R i. Poof. Take the ith ieducible block and suppose fo the sake of contadiction R i C i and R i C i. Define thee sets: fist I = R i C i, second I 1 = R i \ I, and thid J 2 = C i \ I. With this patitioning of the ows and columns, we can pemute the ith ieducible block into the following fom ( I J 2 ) I A A i = 11 A 12. (3.1) I 1 A 21 A 22 Fist note that A 12 O and A 21 O, othewise the block will be educible. Now conside the lage squae submatix consisting of the set of ows I I 1 J 2 and the same set of columns I J 2 I 1 I A 11 A 12 A T 21 A L = I 1 A 21 A 22. (3.2) J 2 A T 12 A T 22 Fo the columns in the set I 1 and the ows in set J 2 to be in a diffeent block fom i, the submatix of A L indexed by the ow set I J 2 and the column set I I 1 should be educible. But the matix ( I I 1 ) I A 11 A T 21 J 2 A T 12 A T 22 (3.3) is the tanspose of A i (see (3.1)), as A 11 = A T 11. Since A i is ieducible, so is its tanspose shown in (3.3). Theefoe, the ows in ow set J 2 and the columns in column set I 1 cannot be in a diffeent block fom the one that contains I. We have established that eithe R i = C i o R i C i =. If R i C i =, all matching nonzeos in this block should be in length 2 pemutation cycles, as the odd pemutation cycles ae ieducible. Theefoe, we have anothe ieducible block j with R j = C i and C j = R i and the poof is completed. Having defined the fine stuctue of the squae block A S, we efine Coollay 3.6 by using that stuctue. As befoe, let R i and C i denote the ows and columns of the ith ieducible block of A S. We will ode the ows of A and then apply that ode in the evese diection to the columns. We fist ode the ows in the hoizontal block. Then, we ode the ows in squae block A S using the fine stuctue as follows. Let i be an ieducible block whose ows ae yet to be odeed. If the column set C i is equal to the ow set R i, ode R i. If C i R i =, ode R i and then the ows coesponding to the columns in C i. Afte this blockwise odeing of all the ows in the squae block A S, we ode the ows in the vetical block. We do not specify the ode of the ows in a subblock it can be abitay. Now applying the ode obtained to the columns in the evese diection esults in a matix that is symmetic along the anti-diagonal. A sample matix odeed with this pocedue is shown in Fig. 3.3. We note that Theoem 3.5, Coollay 3.6, and Lemma 3.7 hold fo stuctually full ank, symmetic matices. In paticula, the fine decomposition of such a matix has squae ieducible blocks with a ow set R i and a column set C i whee eithe R i = C i o thee exists anothe ieducible block j with R j = C i and C j = R i.
BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES 11 8 4 9 1 10 13 15 11 17 20 2 5 19 12 16 18 14 6 7 3 3 7 6 14 18 16 12 19 5 2 20 17 11 15 13 10 1 9 4 8 nz = 52 Fig. 3.3. A 20 20 symmetic matix is shown with the fine decomposition of the squae block. Among the six ieducible blocks of A S, two have the same set of ows and columns, i.e., R i = C i. These appea as the anti-diagonal blocks. The othe fou come in pais of two, whee the pai is located symmetically aound the anti-diagonal. The pemuted matix is symmetic aound the anti-diagonal. REFERENCES [1] P. J. Cameon, Notes on algebaic stuctues. Available at http://www.intute.ac.uk/, last accessed 12 Feb, 2008. [2] I. S. Duff, On pemutations to block tiangula fom, Jounal of the Institute of Mathematics and its Applications, 19 (1977), pp. 339 342. [3] I. S. Duff, A. M. Eisman, and J. K. Reid, Diect Methods fo Spase Matices, Oxfod Univesity Pess, London, 1986. [4] I. S. Duff and S. Palet, Stategies fo scaling and pivoting fo spase symmetic indefinite poblems, SIAM Jounal on Matix Analysis and Applications, 27 (2005), pp. 313 340. [5] I. S. Duff and J. K. Reid, An implementation of Tajan s algoithm fo the block tiangulaization of a matix, ACM Tansactions on Mathematical Softwae, 4 (1978), pp. 137 147. [6] L. Lovasz and M. D. Plumme, Matching Theoy, Noth-Holland mathematics studies, Elsevie Science Publishes, Amstedam, Nethelands, 1986. [7] M. Macus and H. Minc, A Suvey of Matix Theoy and Matix Inequalities, Dove, (Unabidged, unalteed epublication of the coected (1969) pinting of the wok published by Pindle, Webe, & Schmidt, Boston, 1964), 1992. [8] A. Pothen, Spase null bases and maiage theoems, PhD thesis, Depatment of Compute Science, Conell Univesity, Ithaca, New Yok, 1984. [9] A. Pothen and C.-J. Fan, Computing the block tiangula fom of a spase matix, ACM Tansactions on Mathematical Softwae, 16 (1990), pp. 303 324.