Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

BOUNDED RA TIOS OF PR ODUCTS OF PRINCIP AL MINORS OF POSITIVE DEFINITE MA TRICES H. TRACY HALL AND CHARLES R. JOHNSON Abstract. Considered is the m ultiplicative semigroup of r atios of pro ducts of principal minors b ounded o v er all p ositive definite matrices. A long history of literature identifies v arious element s of this semigroup, all of whi c h l i e in a sub- semigroup generated b y Hadamard-Fischer inequalities. Via cone-the oretic tec hniques and the patterns of nullit y among positive semidefinite matrices, a semigroup containing all b ounded ratios is give n. This allows the complete determination of the s emigroup of b ounded ratios for 4-by-4 p ositive definite matrices, whose 46 generators include ratios not i mplied by Hadamard-Fische r and ratios not b ounded by 1. F or n ≥ 5 it is shown that th e cont ainment of semigr oups is stri ct, but a generalization of nullit y patterns, of which one example is giv en, is conject ured to pro vide a finite d etermination of all bounded ratios. 1. Introduction F or an n -by- n matrix A = ( a ij ) a nd an index set S ⊆ N ≡ { 1 , 2 , . . . , n } , A [ S ] is the principal submatrix of A lying in the rows and co lumns indicated by S . Given a co llection α : α 1 , α 2 , . . . , α p ⊆ N of index sets a nd exp onents η 1 , . . . , η p ∈ R + , let α ( A ) = p Y i =1 (det A [ α i ]) η i . A t times we a lso abbreviate α ( A ) a s α η 1 1 · · · α η p p , o mitting a ny exp onents equal to 1. W e are interested in understanding which ratios α β = α ( A ) β ( A ) hav e an upp er b o und that is indep endent of A , pr ovided that A is p ositive def- inite (PD). W e ca ll such ratios b ounde d , and absolutely b oun de d if 1 is an upp er bo und. The Hadamard-Fischer inequalit y [5] provides a classical family of abso- lutely b ounded r atios in which α 1 = S ∪ T and α 2 = S ∩ T , while β 1 = S and β 2 = T , for a ny tw o index sets S, T ⊆ N , with a ll exp onents 1. This classica l family is in fac t bo unded over any clas s of inv ertible matrices with positive prin- cipal minor s a nd weakly sign-s ymmetric non-principa l minors [2], and we will call such r atios Koteljanskii ratios [7]. Histor ical information on determinantal inequal- ities for P D ma tr ices may b e found in ([5], ch. 7 ). Over real symmetric matrices, bo unded ra tio s ar e equiv a lent to multiplicativ e inequalities on the fac e volumes o f a pa rallelotop e. 1991 Mathematics Subje ct Classific ation. Primary 15A45; Seconda ry 15A15, 15A48, 05B35. Key wor ds and phr ases. D eterminantal inequality , b ounded ratio, matrix, posi tiv e definite, posi tive semi definite, principal m i nor, multiplicativ e semigroup, cone of inequalities, rank type, n ullity t yp e, asymptotic nu llity t yp e, matroid, face v olume inequality . 1 2 H. TRA CY HALL AND CHARLES R. JOHNSON A r atio is deter mined, up to equiv a le nce, by the overall ex po nent (n umer ator min us deno mina tor) of each nonempty subset of N . The exp onent of the empty set is chosen to mak e the s um of e xp o nents 0, and the collection of 2 n exp onents is called the fo rmal lo garithm of α β , written log( α β ). F orma l logar ithms span a hyperplane in R (2 N ) orthogo nal to the all-1’s vector e . Let B n be the set o f bounded ratios, A n the set o f absolutely b ounded ratios, and K n the set of pro ducts o f po sitive p ow ers of Koteljanskii ratio s . Then, for all n , K n ⊆ A n ⊆ B n . These are m ultiplicativ e semigroups, and there are corre- sp onding co nes lo g( K n ) ⊆ log( A n ) ⊆ log( B n ). All pr eviously known multiplicativ e determinantal inequalities lie in K n . In [6], nece s sary conditions for b oundedness were giv en, implicitly defining a semigroup E n with B n ⊆ E n , and it was shown that K 3 = E 3 but A 4 6 = E 4 . Here, w e define a new s emigroup D n by imp os- ing linea r inequalities on log( D n ) co ming fro m ra nk deficient po sitive s e midefinite (PSD) matrices. W e sho w that B n ⊆ D n ⊆ E n for all n . W e also show that A 4 6 = B 4 = D 4 6 = E 4 , a nd give explicit generators for B 4 . Going further, w e show that B 5 6 = D 5 , but we conjecture that for ea ch n there exists some finite system of linear inequalities, g e neralizing thos e coming from rank deficient P SD ma tr ices, that is sufficient to characterize log( B n ). 2. Cone theoretic determina tion of D n W e call a ratio α β homo gene ous if α ( D ) /β ( D ) = 1 for every PD diagonal ma trix D , and ca ll the set of a ll such ratios H n . All b ounded ratios are homog eneous [6], and log( H n ) is the subspace of R (2 N ) that is orthogo nal to the following s et of n + 1 vectors: the all-1’s vector e and, for eac h i ∈ N , the v ector whose ent ry T ( as T ranges ov er T ⊆ N ) is 1 whenever i ∈ T , and 0 otherwise . A homog eneous ra tio is uniquely deter mined by the exp onents of subsets with cardinality at lea st 2. Since the dimension of log( K n ) is 2 n − n − 1, the cone lo g ( B n ) ha s full dimension within the subspace log ( H n ). Let M be a matrix with n columns. W e define nul( M ) ∈ R (2 N ) , the nu l lity typ e of M , as the vector of null space dimensions for subsets of the columns of M . (By conv en tion the empt y set o f columns has n ullity 0.) The nullit y type of M a lso g ives the mult iplicity of 0 as an eigenv alue for any principal submatrix of M ∗ M . F o r any n there exist finitely many dis tinct nullit y types (corr esp onding to repre s ent ations of lab eled ma troids on n elements). Our first r esult is that the inner pro duct of the formal log arithm of a ratio with any nullit y type gives a necessa ry condition for b oundedness . Theorem 1. L et α β b e a homo gene ous r atio, let M b e a matr ix with n c olumns , and define a family of m atr ic es A ε , 0 < ε < 1 , by A ε = M ∗ M + εI . Then α β is b ounde d over the family A ε if and only if log ( α β ) T nu l( M ) ≥ 0 . Pr o of. Let s = log( α β ) T nu l( M ), and let r repr esent the sum of the exp onents in α , which is also the sum of the exp onents in β . Let Λ b e the set of all nonzero eigenv a lues of M ∗ M and its principal submatrices, and let λ min and λ max denote resp ectively the minimum and maximum elements of Λ ∪ { 1 } , so that λ min ≤ 1 ≤ λ max . F or any 0 < ε < 1, the matrix A ε is PD, and the quan tit y α β ( A ε ) can be expressed ent irely in terms of the eigen v alues o f principa l submatr ices of A ε . Each of BOUNDED RA TIOS OVER POSITIVE DEFINITE MA TRICES 3 these eig env alues is ε or lies within the rang e λ min + ε to λ max + ε . The total exp onent of ε in the eigen v alue product is exac tly the inner pr o duct log ( α β ) T nu l( M ) = s . The contribution of a ll remaining eig env alues either to the numerator α ( A ε ) or to the denominator β ( A ε ) lie s within the ra nge λ nr min to ( λ max + 1) nr . Thus we have ε s  λ min λ max + 1  nr ≤ α β ( A ε ) ≤ ε s  λ max + 1 λ min  nr . If s < 0 the left ine q uality implies that α β ( A ε ) is unbounded, and if s ≥ 0 the right inequality implies that α β ( A ε ) is b ounded by a co nstant.  Given n , let ν 1 , . . . , ν ℓ be a complete list of the n ullit y types of matrices with n columns. The dual nu l lity semigr oup D n is defined as the set of homogeneous ratios α β such that lo g( α β ) T ν i ≥ 0 for all 1 ≤ i ≤ ℓ . Corollary 2. F or al l n , B n ⊆ D n . Mor e over, if e ach extr eme r ay of log( D n ) r epr esents a b ounde d r atio, then B n = D n . The dual nullit y c one lo g( D n ) is a finite intersection of half-spac es and th us its se t of extreme r ays can (in principle, and sometimes in practice) be explicitly calculated. Some nullit y types ar e redunda nt to this ca lculation: nul( I ) is tr ivial; nu l([ M 1 | 0 ]) − nul( M 1 ⊕ I ) is or thogonal to H n , so one of these may be omitted; and nu l( M 1 ⊕ M 2 ) = n ul( M 1 ⊕ I ) + n ul( I ⊕ M 2 ). Given a matrix M with n columns, let ρ ( M ) be the v ector in R (2 N ) whose en try for eac h T ⊆ N is the rank of the s ubset T of the columns o f M . Since n ul( M ) + ρ ( M ) is the v ector [ | T | ] of cardinalities, whic h is o rthogona l to H n , nul( M ) and − ρ ( M ) are interchangeable for the pur po ses of calculating D n . This fact c a n b e useful co mputatio nally when M ha s low ra nk. Example 1 . Let n = 3, and let ν 1 , . . . , ν 5 be the nullit y types of the matrice s  1 1 1  ,  0 1 1  ,  1 0 1  ,  1 1 0  , and  1 0 1 0 1 1  . Let E 3 be the intersection of the five sets { α β ∈ H 3 : log( α β ) T ν i ≥ 0 } . The six extreme rays o f log ( E 3 ) a re the forma l log arithms of the Ko teljanskii r atios { 12 } ∅ { 1 }{ 2 } , { 13 } ∅ { 1 }{ 3 } , { 23 } ∅ { 2 }{ 3 } , { 123 } { 1 } { 12 }{ 13 } , { 123 } { 2 } { 12 }{ 23 } , and { 123 } { 3 } { 13 }{ 23 } . W e thus have K 3 ⊆ A 3 ⊆ B 3 ⊆ D 3 ⊆ E 3 ⊆ K 3 and in particular B 3 = K 3 , a s was shown in [6]. The symmetr ic group on n elements acts natura lly on R (2 N ) and preser ves e very set we hav e defined. S et co mplemen tation S c = N \ S als o acts na tur ally and preserves log( K n ) a nd log( H n ). Since S ∅ ( A ) = S c N ( A − 1 ) by Jaco bi’s identit y [5], log( A n ) and log( B n ) are also in v ar iant under the action of complemen tation. T o show that log( D n ) is inv ar ia nt under complementation r equires s o me matr oid the- ory [9]: the complement o f a nullit y type is equiv a lent , up to vectors or thogonal to log( H n ), to the nu llity type of the dual ma troid. Theorem 1 generalizes the kno wn set-theor etic necessa ry conditions ST0, ST1 , and ST2. Theorem 3. [6 ] F or any b ounde d r atio α β and any given index set S ⊆ N , t he sum of exp onent s in α β for sup ersets of S is nonne gative (ST1), and likewise for subsets of S (ST2). Any r atio satisfying ST1 and ST2 also satisfies homo geneity ( S T0). 4 H. TRA CY HALL AND CHARLES R. JOHNSON W e define E n as the set o f r atios sa tisfying the elementary necessa ry c o nditions ST1 and ST2 for every S ⊆ N . These conditions can be restated in terms of nullit y t yp es. Given an index set S ⊆ N with | S | ≥ 3 , define M S to b e the matrix of size ( n − 1) × n that is a p er m utation of [ I | e ] ⊕ I , with the [ I | e ] summand o ccurr ing in the c olumns of S and the I s ummand o ccurr ing in the complement o f S . Then for T ⊆ N , entry T of nu l( M S ) is 1 if S ⊆ T a nd 0 otherwise. Given an index s et S ⊆ N with | S | ≤ n − 2, define M S to b e the matrix o f size 1 × n whose en tries are 0 in the columns of S a nd 1 elsewhere. The n for T ⊆ N , entry T of ρ ( M S ) is 0 if T ⊆ S and 1 otherwise. F or ex a mple, with n = 3, the five matrices listed in Example 1 are M ∅ , M { 1 } , M { 2 } , M { 3 } , and M { 123 } . Prop ositio n 4. A homo gene ous r atio α β satisfies c onditions ST1 and ST2 if and only if log ( α β ) T nu l( M S ) ≥ 0 for | S | ≥ 3 and log ( α β ) T nu l( M S ) ≥ 0 for | S | ≤ n − 2 . Pr o of. The cases | S | ≤ 1 for ST1 and | S | ≥ n − 1 for ST2 are equiv alent to the assumed homogeneity . The ca se ST1 with | S | = 2 will follow from ST2 with | S | = n − 2: for a homogeneous ra tio, ST1 is satisfied for { i , j } if and only if ST2 is satisfied for { i, j } c , as can be seen b y partitioning 2 N int o four gro ups depe nding on the mem b ership o f i and j , and comparing the total exp onents of these groups under ho mogeneity . F o r | S | ≥ 3, log( α β ) T nu l( M S ) is no nneg ative if and only if S occurs a s a subset in α with a total exp onent at least as g reat as in β . F o r | S | ≤ n − 2, log( α β ) T  e − ρ ( M S )  is nonnegative if and only if S o ccurs as a supers e t in α with a tota l expo nent a t least as gr eat as in β . Since α β is homogeneous and nul( M S ) + ρ ( M S ) − e is or thogonal to log( H n ), this is equiv alent to log ( α β ) T nu l( M S ) ≥ 0.  Corollary 5. F or al l n , D n ⊆ E n . The name E 3 in E xample 1 is now justified, and we have s hown, as in [6], that K 3 = E 3 . Since the vectors nul( M S ) for | S | ≤ n − 2 and the vectors o rthogona l to log( H 3 ) to gether span R (2 N ) , log ( B n ) is a p ointed cone: no nontrivial ratio is b oth bo unded and b ounded awa y from 0 . 3. Bounding ra tios not in K n Given a ratio α β in D n \ K n , it may b e difficult to deter mine whether α β is b ounded. W e introduce a technique that can so metimes prove b oundedness, starting with a known o bserv a tion whose first nontrivial case is n = 3: Lemma 6. [4] L et A = ( a ij ) b e an n -by- n PD matrix with inverse B = ( b ij ) . Then for e ach i = 1 , . . . , n (3.1) 2 p a ii b ii + ( n − 2) ≤ n X j =1 p a j j b j j . Eliminating a copy o f i on ea ch side, repla cing the sum with ( n − 1 ) times the maximum, and us ing the fact that b ii = { i } c N ( A ), we c onclude: Corollary 7. F or n ≥ 3 and A PD, and for any i ∈ N , min j 6 = i  { i } { i } c { j }{ j } c ( A )  ≤ ( n − 1) 2 . BOUNDED RA TIOS OVER POSITIVE DEFINITE MA TRICES 5 Example 2 . In [6] it is shown that A 4 6 = E 4 using a p ermutation of R 1 = { 124 } { 134 } { 23 } { 1 } { 4 } { 12 } { 13 } { 14 } { 24 } { 34 } . Observe that this ratio is unchanged when indices 2 and 3 a re interc hanged; we may thus a ssume without loss of generality that { 1 }{ 23 } { 2 }{ 13 } ( A ) ≤ { 1 }{ 23 } { 3 }{ 12 } ( A ). By applying Corolla ry 7 to A [ { 123 } ] with i = 1 we then obtain, fac to ring out a pair of Koteljanskii r atios, R 1 = { 124 } { 2 } { 12 } { 24 } · { 134 } { 4 } { 14 } { 34 } · { 1 } { 23 } { 2 } { 13 } ≤ 4 , which shows that R 1 belo ngs to B 4 . 4. The extreme ra ys of log ( B 4 ) Let µ 1 , . . . , µ 23 represent the distinct nullit y t yp es of column p ermutations of the following seven matr ices: M ∅ , M { 1 } , M { 12 } , M { 123 } , M { 1234 } , M 6 =  1 0 1 1 0 1 1 1  , and M 7 =  1 1 1 1 0 1 2 3  . The in ter section of H 4 with th e 23 half-spaces { x ∈ R 16 : x T µ i ≥ 0 } con tains D 4 and thus B 4 . There are v ar ious s oftw ar e pack ages av ailable for computing the ex- treme rays of a n intersection of half-spaces in exact arithmetic; in this ca se the calcu- lation was done with custom- w r itten so ftw are, whose results w ere later verified with the program lrs [1] (av ailable at ht tp:// cgm.cs .mcgill.ca/ ~ avis/C /lrs. html ). The 46 extreme rays include the 24 extreme rays of lo g( K 4 ) (those with | S | = | T | = | S ∩ T | + 1) and the 6 p er m utations of log( R 1 ) from Example 2. The r atios co r re- sp onding to the rema ining 1 6 ex tr eme rays ar e p ermutations of (4.1) R 2 = { 1234 } { 1 234 } { 23 } { 24 } { 34 } { 1 } ∅ { 123 } { 124 } { 1 34 } { 23 4 } { 2 } { 3 } { 4 } , (4.2) R 3 = { 1234 } { 23 } { 24 } { 34 } { 1 } { 1 } ∅ { 123 } { 12 } { 13 } { 14 } { 2 } { 3 } { 4 } , and their complemen ts. Since R 2 and R 3 are b oth symmetric in indices 2 and 3, we ma y make the same a ssumption as in Exa mple 2 and conclude from Co rollar y 7 that R 2 = { 1234 } { 24 } { 124 } { 234 } · { 1234 } { 13 } { 123 } { 134 } · { 34 } ∅ { 3 } { 4 } · { 1 } { 23 } { 2 } { 13 } ≤ 4 and R 3 = { 1234 } { 24 } { 124 } { 234 } · { 124 } { 1 } { 12 } { 14 } · { 34 } ∅ { 3 } { 4 } · { 1 } { 23 } { 2 } { 13 } ≤ 4 . Although the proven b ound is 4, e x pe r imental evidence sugge sts that the ra tios R 2 and R 3 are abs o lutely b ounded, which would imply K 4 6 = A 4 , and that the c o rrect upper b ound for R 1 is 27 16 , which v alue it is known to appr oach [8]. This is the first known case showing A n 6 = B n for any n . It has b e en shown [3] that b ounded ratios on M- ma trices or inverse M-ma trices are absolutely bounded, and preliminary work shows the sa me for (inv ertible) totally nonnegative ma trices as far a s n < 6. The listed nullit y types a r e thus sufficient to define D 4 , and indeed B 4 . 6 H. TRA CY HALL AND CHARLES R. JOHNSON Theorem 8. The semigr oup B 4 is gener ate d, up to p ermut ation and c omplemen- tation, by p ositive p owers of { 12 } ∅ { 1 }{ 2 } , { 123 }{ 1 } { 12 }{ 13 } , R 1 , R 2 , and R 3 . Theorem 9. The sets B 4 and D 4 ar e e qual: a r atio α β ∈ H 4 is b ounde d if and only if log ( α β ) T µ i ≥ 0 for al l i = 1 , . . . , 23 , with µ i as liste d ab ove. W e end the section with an exa mple showing that D 4 6 = E 4 . The ratio α β = { 1234 } { 134 } { 12 } { 14 } { 23 } { 24 } { 3 } ∅ { 123 } { 124 } { 234 } { 13 } { 34 } { 1 } { 2 } { 4 } satisfies ST1 and ST2 with resp ect to every S ⊆ { 1 , 2 , 3 , 4 } . How ever, the inner pro duct lo g( α β ) T nu l( M 6 ) g ives − 1, and α β is not b ounded. 5. The case n = 5 A ratio in the se migroup D n is b ounded nea r ε = 0 for any matrix family of the form ( B + εC ) ∗ ( B + εC ), where B is singular and C is generic in the sense that C x = B y and B x = 0 imply x = 0 . W e ha ve shown that for n = 4 such matrix families, whose b ehavior depends only on the nullit y type of B , a re sufficient to determine B 4 . The following example shows that for n ≥ 5 , more genera l matrix families must b e co nsidered. Example 3 . Let n = 5 and consider the ratio Q = { 1234 5 } { 1345 } { 234 5 } { 123 } { 125 } { 3 4 } { 4 5 } { 1 } { 2 } ∅ { 1234 } { 123 5 } { 14 5 } { 245 } { 345 } { 1 2 } { 1 3 } { 23 } { 4 } { 5 } . W e claim that Q ∈ D 5 . Since Q is se lf- c o mplement ary , it is only necessar y to chec k against ma trices of rank up to ⌊ 5 2 ⌋ , i.e. M o f size 2 × 5. Uniformly deleting a sp ecified index element in the sets of Q yields a pr o duct of K oteljanskii ra tios in all five case s , which eliminates the ca se in which M has a zero co lumn (using homogene ity). The nu llity t ype of M now depe nds only on a pa rtition of the columns { 1 , 2 , 3 , 4 , 5 } in to parallel classes . Partitions into ex a ctly tw o blo cks give the nullit y type of a direct sum and a re thus redundant, leaving 37 cases to chec k. Now let P =       1 1 1 1 1 0 ε 0 1 2 0 0 ε 1 2 0 0 0 ε 0 0 0 0 0 ε       , define A ε as P ∗ P , and consider the p olyno mial de t A ε [ S ] for some S ⊆ N . This is a nonnegative function o f ε , and so for every S the do minating term for ε small m ust be an even p ow er C S ε 2 d S for C S > 0 and d S a nonnegative in teger. Ca ll the vector [ d S ] ∈ R (2 N ) the asymptotic nu l lity t yp e of P , or as n( P ). The fact that log( Q ) T asn( P ) = − 1 means tha t the ratio Q is not b ounded, and B 5 6 = D 5 . W e ex pec t that a ny o ne-parameter matrix family making a particular ratio un- bo unded can b e adequately a pproximated by p olynomia ls in ε , and further that the maximum necess ary degree o f the approximation dep ends only on n . W e thus exp ect log( B n ) to be a p olyhedr al co ne for all n , as it is for n ≤ 4. BOUNDED RA TIOS OVER POSITIVE DEFINITE MA TRICES 7 Conjecture 1 . Given any n , define the asymptotic n ul lity typ e asn( P ) of an in- vertible n × n matrix P of p olynomials in ε as in Exampl e 3. Then ther e ex ists a finite list of p olynomial matric es P 1 , . . . , P ℓ such that a homo gene ous r atio α β is b ounde d if and only if log( α β ) T asn( P i ) ≥ 0 for al l 1 ≤ i ≤ ℓ . References 1. David Avis, lrs: A r ev ise d implementation of the r everse sea r c h vert ex enumer ation algo- rithm , Polyto p es—com binatorics and computation (Oberwolfach , 1997), DMV Sem., v ol. 29, Birkh¨ auser, Basel, 2000, pp. 177–198 . 2. David Carlson, We akly sign-symmetri c matric es and some determinantal ine q ualities , Colloq. Math. 17 (1967) , 123–129. 3. Shaun F allat, H. T r acy Hall, an d Charles R. Jo hnson, Char acteri zati on of pr o duct ine qualities for princi p al minors of m-matric es and inverse m- matric es , Quart. J. Math. Oxford Ser. (2) 49 (1998), 451–458. 4. Mir oslav Fiedler, Rela tions b etwe en the diagonal elements of two mutual ly inverse p ositiv e definite matrices , Czec hoslov ak M ath. J. 14 (8 9) (196 4), 39–51. 5. Roger A. Horn and Charles R. Johnson, Matrix analysis , Cambridge Universit y Press, Cam- bridge, 1985. 6. Charles R. Johnson and W a yne W. Barrett, Determinantal ine qualities for p ositive definite matric es , Discrete Math. 119 (1993), no. 1-3, 97–106, ARIDAM IV and V (New Brunswick, NJ, 1988/198 9). 7. David M. Kote ljanskii, The the ory of nonne gative and oscil lating matrices , Amer. Math . So c. T r ansl. (2) 27 (1963) , 1–8. 8. Peter McNamara, Ra tios of minors of p ositive semidefinite matric es , Researc h Experi ences for Undergraduates Pr ogram, College of William and Mary (Summer 1997), Advisor: H. T racy Hall. 9. James G. Oxley , Matr oid the ory , Oxford Science Publi cations, The Clarendon Press Oxf ord Unive rsity Press, New Y ork, 1992. Dep ar tment of Ma thema tics, Brigham Young University, Pro vo, UT 8 4602 E-mail addr ess : h.tracy@gmail .com Dep ar tment of M a them a tics, College of William and Mar y, Williamsburg, V A 23185 E-mail addr ess : crjohnso@math .wm.edu