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1 RANGES OF SYLVESTER MAPS AND A MNMAL RANK PROBLEM ANDRE CM RAN AND LEBA RODMAN Abstract t is proved that the range of a Sylvester map defined by two matrices of sizes p p and q q, respectively, plus matrices whose ranks are bounded above, cover all p q matrices The best possible upper bound on the ranks is found in many cases An application is made to a minimal rank problem that is motivated by the theory of minimal factorizations of rational matrix functions Key words Sylvester maps, nvariant subspaces, Rank AMS subject classifications 15A6, 15A99 1 ntroduction Let F be a commutative field We let F p q stand for the set of p q matrices with entries in F; F p 1 will be abbreviated to F p The following minimal rank problem was stated in [8, Section 6] for the case when F is the complex field C: Problem 11 Given A F n n, and given an A-invariant subspace M F n, find the smallest possible rank, call it µa,m, for the difference A Z, where Z runs over the set of all n n matrices with entries in F for which there is a Z- invariant subspace N F n complementary to M Also, find structural properties, or description, of such matrices Z The problem for F = C is intimately connected with minimal factorizations of rational matrix functions, in particular, if certain additional symmetry properties of A, M, and Z are assumed; see [8] for more details Pairs of matrices A,Z that have a pair of complementary subspaces M, N, of which the first is A-invariant and the second is Z-invariant, but without explicit rank conditions on A Z, are studied in [1, 2], for example, in connection with complete minimal factorization of rational matrix functions Received by the editors October 12, 29 Accepted for publication January 2, 21 Handling Editor: Harm Bart Afdeling Wiskunde, Faculteit der Exacte Wetenschappen, Vrije Universiteit Amsterdam, De Boelelaan 181a, 181 HV Amsterdam, The Netherlands ran@csvunl College of William and Mary, Department of Mathematics, POBox 8795, Williamsburg, VA , USA lxrodm@mathwmedu The research of this author was partially supported by the European Network for Analysis and Operator Theory, and by College of William and Mary Faculty Research Assignment and Plumeri Award The work was completed during this author s visit at the Afdeling Wiskunde, Vrije Universiteit, whose hospitality is gratefully acknowledged 126

2 Ranges of Sylvester Maps and a Minimal Rank Problem 127 n the general formulation, Problem 11 appears to be difficult, even intractable, especially the part concerning properties or description of all matrices Z To illustrate, assume F is algebraically closed, and let A = 1, M = Span 1 x w Then µa,m = 1, and a matrix Z =, x,y,z,w F, has the properties y z that ranka Z = 1 and Z has an invariant subspace complemented to M if and only if Z is not a nonzero scalar multiple of A and the equality xz + y1 w = holds f A and M are as in Problem 11, by applying a similarity transformation we can assume without loss of generality that M is spanned by first p unit coordinate vectors in F n ; thus A has the block form A1 A A = 12, A 2 where A 1 F p p, A 2 F n p n p f the minimal polynomials of A 1 and A 2 are coprime, then it is easy to see that µa,m =, ie, A has an invariant subspace N Q complementary to M ndeed, such N is spanned by the columns of, where the matrix Q satisfies the equation A1 A 12 A 2 or Q = Q A 2, QA 2 A 1 Q = A t is well known that the Sylvester map Q QA 2 A 1 Q is invertible if and only if the minimal polynomials of A 1 and A 2 are coprime See, eg, [5, 7] for this fact; although this was established in [5, 7] only for the complex field, the extension to any algebraicallyclosed field isimmediate, and toprovethis fact forthe generalfield Fone considersthe algebraicclosureoff Hence 11 can be solvedfor Q forany givena 12, and µa, M = is established This example shows close connections of Problem 11 with properties of Sylvester maps There is a large literature in mathematical and engineering journals on numerical analysis involving Sylvester maps; see, eg, [3, 4] and the references cited there Connected to the Sylvester map the following problem appears in the theory of control for coordination see [6] Given is a linear system ẋt = Axt, where the

3 128 ACM Ran and L Rodman matrix A has the form A 11 A 1c A = A 22 A 2c A cc with respect to a fixed decomposition X = X 1 +X 2 +X c of the state space One allows transformations A S 1 AS, where S is of the form S 1 S = S 2 Then A 11 A 11 S 1 S 1 A cc +A 1c S 1 AS = A 22 A 22 S 2 S 2 A cc +A 2c A cc From the point of view of communicating as little as possible between the coordinator acting in X c and the subsystems acting in X 1 and X 2, it is of interest to study when the ranks of A ii S i S i A cc +A ic are as small as possible for i = 1,2 t is precisely this problem we shall discuss in the next section 2 Ranges of Sylvester maps We recall the definition of invariant polynomials For A F p p, we let λ A = Eλdiag φ A,1 λ,,φ A,p λfλ, where Eλ, Fλ are everywhere invertible matrix polynomials, and φ A,j are scalar monic polynomials such that φ A,j is divisible by φ A,j+1, for j = 1,,p 1 The polynomials φ A,j are called the invariant polynomials of A; φ A,1 is in fact the minimal polynomial of A For two matrices A 1 and A 2 over the field F of sizes p p and q q, respectively, define the nonnegative integer sa 1,A 2 as max{j 1 j min{p,q}, φ A1,jλ and φ A2,jλ are not coprime} The maximum of the empty set in this formula is assumed to be zero Clearly, sa 1,A 2 = if and only if the minimal polynomials of A 1 and A 2 are coprime f all eigenvalues of A 1 and A 2 are in F in particular if F is algebraically closed, then sa 1,A 2 = max λ F min{dimkera 1 λ,dimkera 2 λ}

4 Ranges of Sylvester Maps and a Minimal Rank Problem 129 Consider the linear Sylvester map for fixed A 1 and A 2 T : F p q F p q defined by TS = SA 2 A 1 S, S F p q, Theorem 21 a Every matrix X F p q can be written in the form X = TS+Y, for some S F p q and some Y F p q with ranky sa 1,A 2 b Assume that sa 1,A 2, and that the greatest common divisor of the minimal polynomials of A 1 and A 2 have all their roots in F in particular, this condition is always satisfied if F is algebraically closed Then for fixed A 1 and A 2, there is a Zariski open nonempty set Ω of F p q such that for every X Ω, there is no representation of X in the form X = TS+Y, where S,Y F p q are such that ranky < sa 1,A 2 Theorem 21 can be thought of as a generalization of the well known fact that the Sylvester map is a bijection if and only if the minimal polynomials of A 1 and A 2 are coprime Consider the following example to illustrate Theorem 21 Let A 1 = The invariant polynomials are 1 and A 2 = φ A1,1λ = φ A2,1λ = λλ 1, 1 1 φ A1,2λ = λ, φ A1,2λ = λ 1, φ A1,3λ = φ A2,3λ = 1 We have sa 1,A 2 = 1 The range of the Sylvester map is easy to find: RangeT =,

5 13 ACM Ran and L Rodman where by we denote arbitrary entries which are independent free variables For every X = [x i,j ] F 3 3, we have X = x 1,1 x 2,1 1 [ ] 1 x 3,2 x 3,3 + 3 Proof of Theorem 21 Part a We use the rational canonical forms for A 1 and A 2 see, eg, [5], together with the invariance ofthe statement of the theorem and of its conclusions under similarity transformations A 1 G 1 1 A 1G 1, A 2 G 1 2 A 2G 2, G 1 and G 2 invertible where We may assume therefore without loss of generality that A 1 = diaga 1 1,,Au 1 and A 2 = diaga 1 2,,Av 2, 31 A j 1 = diaga j,1 1,,A j,γ1,j 1, j = 1,,u, and where the characteristic polynomials of A j,k 1, k = 1,,γ 1,j, are all powers of the same monic irreducible polynomial f 1,j The matrices A j,k 1 are nonderogatory, ie, the minimal and the characteristic polynomials of A j,k 1 coincide n addition, we require that the irreducible polynomials f 1,1,,f 1,u are all distinct Similarly, A j 2 = diaga j,1 2,,A j,γ2,j 2, j = 1,,v, where the characteristic polynomials of A j,k 2, k = 1,,γ 2,j, are all powers of the same monic irreducible polynomial f 2,j, and the polynomials f 2,1,,f 2,v are all distinct Again, the matrices A j,k 2 are nonderogatory Moreover, we arrange the blocks A 1 1,,Au 1 and A 1 2,,Av 2 so that but the irreducible polynomials f 1,1 = f 2,1,,f 1,l = f 2,l, f 1,1,,f 1,u,f 2,l+1,,f 2,v are all distinct The case when l =, ie, the polynomials f 1,j and f 2,j are all distinct, is not excluded; in this case the subsequent arguments should be modified in

6 Ranges of Sylvester Maps and a Minimal Rank Problem 131 obvious ways Note that since powers of distinct irreducible polynomials are coprime, it follows that the characteristic polynomials of the matrices are pairwise coprime A 1 1,,Au 1,Al+1 2,,A v 2 32 We assume in addition that A j,k 1 are companion matrices To set up notation, we let e j be a row with 1 in the jth position and zeros in all other positions the number of components in e j will be evident from context, and analogously let e T j the transpose of e j be the column with 1 in the jth position and zeros in all other positions Let ξ 1,j,k resp, ξ 2,j,k be the size of the matrix A j,k 1 resp, A j,k 2 Thus, we let A j,k 1 = e 2 e 3 e ξ1,j,k α 1,j,k 33 or [ A j,k 1 = e T 2 e T 3 e T ξ 1,j,k α T 1,j,k ] 34 for some row α 1,j,k with entries in F, and analogously, A j,k 2 = e 2 e 3 e ξ2,j,k α 2,j,k 35 or [ A j,k 2 = e T 2 e T 3 e T ξ 2,j,k α T 2,j,k ] 36 for some row α 2,j,k The forms 33 and 35 will be used if γ 1,j γ 2,j, and the forms 34 and 36 will be used if γ 1,j > γ 2,j We return to the Sylvester map T Conformably with 31, we partition S = [S j1,j 2 ] j1=1,,u;j 2=1,,v

7 132 ACM Ran and L Rodman Thus, TS = SA 2 A 1 S = [S j1,j 2 A j2 2 A j1 1 S j1,j 2 ] j1=1,,u;j 2=1,,v Also, if X F p q is an arbitrary matrix, then we partition again conformably with 31: X = [X j1,j 2 ] j1=1,,u;j 2=1,,v We will show that for any given X F p q, there exist S j,j F ξ1,j,1+ +ξ1,j,γ 1,j ξ2,j,1+ +ξ2,j,γ 2,j, j = 1,,l, with the property that X j,j = Y j,j + S j,j A j 2 A j 1 S j,j, 37 for some matrix Y j,j such that ranky j,j min{γ 1,j,γ 2,j }, j = 1,,l 38 Assuming that we have already shown the existence of S j,j satisfying 38 and 37, we can easily complete the proof of Part a ndeed, let and notice that µ = sa 1,A 2 Now let µ = max j=1,,l min{γ 1,j,γ 2,j }, Y j,j = W j,j Z j,j, j = 1,,l be a rank decomposition, where the matrix W j,j resp, Z j,j has µ columns resp, µ rows We also put formally W j,j = for j = l + 1,,u, and and Z j,j = for j = l+1,,v Using the property that the characteristic polynomials of matrices 32 are pairwise coprime, and that the Sylvester map S SB 2 B 1 S is onto if the characteristic polynomials of B 1 and B 2 are coprime, we find for S j1,j 2 F ξ1,j 1,1+ +ξ1,j 1,γ 1,j 1 ξ 2,j2,1+ +ξ 2,j2,γ 2,j2 j 1 = 1,,u, j 2 = 1,,v, j 1,j 2 {1,1,l,l}, such that X j1,j 2 = W j1,j 1 Z j2,j 2 + S j1,j 2 A j2 2 A j1 1 S j1,j 2

8 Ranges of Sylvester Maps and a Minimal Rank Problem 133 Now letting Y = W 1,1 W 2,2 W u,u [ ] Z1,1 Z 2,2 Z v,v, we have X = TS+Y, and obviously, ranky µ Thus, it remains to show the existence of S j,j satisfying 37 and 38 We fix j, j = 1,,l We assume that γ 1,j γ 2,j, thus 33 and 35 will be used; if γ 1,j > γ 2,j the proof is completely analogous using 34 and 36 Choose rows α 1,j,k k = 1,,γ 1,j so that the characteristic polynomials of the matrices B j,k 1 := are coprime to f 2,j = f 1,j, and let B j 1 = diag Therefore, we can find S j,j so that e 2 e 3 e ξ1,j,k α 1,j,k, k = 1,,γ 1,j, B j,1 1,,B j,γ1,j 1 X j,j = S j,j A j 2 B j 1 S j,j Now 37 holds with Y j,j = B j 1 A j 1 S j,j, and since the matrix Y j,j has at most γ 1,j nonzero rows, we have ranky j,j γ 1,j, as required Part b We assume that A 1 and A 2 have the form 31, and use the notation introduced in the proof of Part a We have l 1 Let j be such that µ = min{γ 1,j,γ 2,j } Without loss of generality we may assume j = 1 Let p 1 p 1 and q 1 q 1, be the size of A 1 1 = diaga 1,1 1,,A 1,γ1,1 1 and A 1 2 = diaga 1,1 2,,A 1,γ2,1 2, respectively t is easy to see that it suffices to find a Zariski open nonempty set Ω 1 of F p1 q1 such that for every X 1 Ω 1, there is no representation of X 1 in the form X 1 =

9 134 ACM Ran and L Rodman SA 1 2 A 1 1 S+Y, where S,Y Fp1 q1 and ranky < min{γ 1,1,γ 2,1 } Because of the hypotheses of Part b, we may assume that every matrix A 1,w k, w = 1,2,,γ k,1, k = 1,2, is an upper triangular Jordan block with the same eigenvalue λ; let the size of this block be p k,w p k,w Consider a matrix of the form SA 1 2 A1 1 S, S Fp1 q1, which is partitioned where the block Q α,β has the size size ofa 1,α 1 SA 1 2 A 1 1 S = [Q γ1,1, γ2,1 α,β] α=1, β=1, 39 size ofa 1,β 2 Since the A 1,w k s are Jordan blocks, the bottom left corners of the blocks Q α,β are all zeros Now partition γ1,1, γ2,1 X 1 = [X α,β ] α=1, β=1 Fp1 q1 comformably with the right hand side of 39 The Zariski open set Ω 1 consists of exactly those matriceds X 1 for which the γ 1,1 γ 2,1 matrix formed by the bottom left corners of the X α,β s has the full rank, equal to min{γ 1,1,γ 2,1 } 4 A special case of the minimal rank problem Given a subspace M F n and a matrixz F n n, we saythat M is a complementary Z-invariant subspace if M is Z-invariant and some direct complement to M in F n is also Z-invariant Denote by CM the set of all matrices Z for which M is a complementary invariant subspace The following problem is closely related to Problem 11 Problem 41 Given a matrix A F n n and its invariant subspace M F n, find the smallest possible rank of the differences A Y, where Y is an arbitrary matrix in CM, and find a matrix Z CM such that the difference A Z has this smallest possible rank n fact, Problem 41 requires an extra condition for Z in comparison with Problem 11, namely, that M is an invariant subspace for Z Using similarity, we assume without loss of generality that { } x M = : x F p A1 A 12, A =, A 1 F p p, A 2 F q q A 2 Theorem 21 sheds some light on Problem 41, as follows Theorem 42 Let κ := sa 1,A 2 There exists a matrix Z CM such that ranka Z κ 41

10 Ranges of Sylvester Maps and a Minimal Rank Problem 135 Moreover, Z can be taken in the form A1 W Z = A 2 for some W 42 Proof We use Theorem 21 ndeed, if Z is in the form 42, then Z CM Q if and only if for some matrix Q F p q the subspace Span is Z-invariant, ie, the equation holds, or equivalently, A1 W A 2 Q = Q QA 2 A 1 Q W A 12 = A 12 ByTheorem21,suchQexistsforsomeW withthe propertythat ranka 12 W κ Since obviously the result follows ranka Z = ranka 12 W, A 2 REFERENCES [1] H Bart and H Hoogland Complementary triangular forms of pairs of matrices, realizations with prescribed main matrices, and complete factorization of rational matrix functions Linear Algebra Appl, 13: , 1988 [2] H Bart and RA Zuidwijk Simultaneous reduction to triangular forms after extension with zeroes Linear Algebra Appl, 281:15 135, 1998 [3] R Bhatia and P Rosenthal How and why to solve the operator equation AX XB = Y Bull London Math Soc, 291:1 21, 1997 [4] K Datta The matrix equation XA BX = R and its applications Linear Algebra Appl, 19:91 15, 1988 [5] FR Gantmacher The Theory of Matrices Chelsea Publishing Co, New York, 1959 [6] PL Kempker, ACM Ran, and JH van Schuppen Construction of a coordinator for coordinated linear systems Proceedings of the European Control Conference 29, 29 [7] P Lancaster and M Tismenetsky The Theory of Matrices With Applications, 2nd edition Academic Press, Orlando, 1985 [8] L Lerer, MA Petersen, and ACM Ran Existence of minimal nonsquare J-symmetric factorizations for self-adjoint rational matrix functions Linear Algebra Appl, 379: , 24