Robustness and Perturbations of Minimal Bases

Transcription

1 Robustness an Perturbations of Minimal Bases Paul Van Dooren an Froilán M Dopico December 9, 2016 Abstract Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory, linear systems theory, an coing theory It is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix an to call such matrix simply a minimal basis Very recently, minimal bases, as well as the closely relate pairs of ual minimal bases, have been applie to a number of problems that inclue the solution of general inverse eigenstructure problems for polynomial matrices, the evelopment of new classes of linearizations an l-ifications of polynomial matrices, an backwar error analyses of complete polynomial eigenstructure problems solve via a wie class of strong linearizations These new applications have reveale that although the algebraic properties of minimal bases are rather well unerstoo, their robustness an the behavior of the corresponing ual minimal bases uner perturbations have not yet been explore in the literature, as far as we know Therefore, the main purpose of this paper is to stuy in etail when a minimal basis M(λ) is robust uner perturbations, ie, when all the polynomial matrices in a neighborhoo of M(λ) are minimal bases, an, in this case, how perturbations of M(λ) change its ual minimal bases In orer to stuy such problems, a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conitions is introuce an, base on it, we prove that polynomial matrices are generically minimal bases with very specific properties In aition, some applications of the results of this paper are iscusse Key wors backwar error analysis, ual minimal bases, genericity, linearizations, l-ifications, minimal bases, minimal inices, perturbations, polynomial matrices, robustness, Sylvester matrices AMS subject classification 15A54, 15A60, 15B05, 65F15, 65F35, 93B18 1 Introuction Minimal bases, forme by vectors with polynomial entries, of a rational vector subspace were mae popular in stanar references of control theory an linear systems theory as those of Wolovich [23], Forney [13], an Kailath [17], although all three of them cite earlier work for some theoretical evelopments on the so-calle minimal polynomial bases For instance, one can rea in [17, p 460] the following sentence: IC Gohberg pointe out to the author that the significance of minimal bases was perhaps first realize by J Plemelj in 1908 an then substantially evelope in 1943 by NI Mushkelishvili an NP Vekua This means that this paper eals with an almost 110 years ol classical mathematical notion However, the iscovery of the importance of this concept in applications ha to wait until the 1970s, when the contributions of authors such as Wolovich, Department of Mathematical Engineering, Université catholique e Louvain, Avenue Georges Lemaître 4, B-1348 Louvain-la-Neuve, Belgium paulvanooren@uclouvainbe Supporte by the Belgian network DYSCO (Dynamical Systems, Control, an Optimization), fune by the Interuniversity Attraction Poles Programme initiate by the Belgian Science Policy Office Departamento e Matemáticas, Universia Carlos III e Mari, Avenia e la Universia 30, 28911, Leganés, Spain opico@mathuc3mes Supporte by Ministerio e Economía, Inustria y Competitivia of Spain an Fono Europeo e Desarrollo Regional (FEDER) of EU through grants MTM REDT, MTM P (MINECO/FEDER, UE) 1

2 Forney, Kailath, an others, provie computational schemes for constructing a minimal basis from an arbitrary polynomial basis, an showe the key role that this notion plays in multivariable linear systems These systems coul be moele by rational matrices, polynomial matrices, or linearize state-space moels, an ha tremenous potential for solving analysis an esign problems in control theory as well as in coing theory The reaer is referre to [13, 17] an the references therein for more information on minimal bases an their applications, an also to the brief revision inclue in the Section 2 of this paper Moreover, many papers have been publishe after [17] on the computation an applications of minimal bases an some of them can be foun in the references inclue in [3] Very recently, minimal bases, an the closely relate notion of pairs of ual minimal bases, have been applie to the solution of some problems that have attracte the attention of many researchers in the last fifteen years For instance, minimal bases have been use (1) in the solution of inverse complete eigenstructure problems for polynomial matrices (see [7, 8] an the references therein), (2) in the evelopment of new classes of linearizations an l-ifications of polynomial matrices [5, 9, 10, 11, 12, 18, 22], which has allowe to recognize that many important linearizations commonly use in the literature are constructe via ual minimal bases (incluing the classical Frobenius companion forms), (3) in the explicit construction of linearizations of rational matrices [1], an (4) in the backwar error analysis of complete polynomial eigenvalue problems solve via the so-calle block Kronecker linearizations of polynomial matrices [10, Section 6], which inclue the interesting class of Fieler linearizations (see [5, 10] for references on this class of linearizations), but o not inclue most of the linearizations an l-ifications that can be constructe from minimal bases [5, 9, 10, 11, 12, 18, 22] See also [19] for aitional references on the role playe by ual minimal bases in backwar error analyses The backwar error analysis in [10, Section 6] uses heavily the following two key ieas: (1) that the very particular minimal bases of egree one which are blocks of the block Kronecker linearizations are robust uner perturbations, in the sense that all the polynomial matrices in a neighborhoo of these minimal bases are also minimal bases with similar properties; an (2) that if these particular minimal bases are perturbe by a certain magnitue, then a particular choice of their ual minimal bases changes essentially by the same magnitue These results have mae clear that for extening the backwar error analysis in [10, Section 6] to much more general contexts that inclue many other classes of linearizations an also l-ifications of polynomial matrices, it is necessary to stuy in epth the robustness of general minimal bases an the behaviour of their ual minimal bases uner perturbations, questions that have not been consiere so far in the literature To solve the robustness an perturbation problems for minimal bases iscusse in the previous paragraph is the main goal of this paper, which is achieve in Sections 6 an 7 We emphasize that the solution of these problems is base on a number of aitional results that, in our opinion, are by themselves interesting contributions to the theory an applications of minimal bases For instance, a new characterization of minimal bases in terms of Sylvester matrices is presente in Section 3, an we prove in Section 5 that polynomial matrices are generically minimal bases with very specific properties that are encoe in the concept of polynomial matrices with full-sylvesterrank, which are stuie in Section 4 In Section 8, the stanar rank characterization of minimal bases is connecte to the results in this paper This work is complete with a iscussion of some preliminary applications in Section 9 an the conclusions are presente in Section 10 2 Preliminaries The results in Sections 2, 3, an 4 of this paper hol for an arbitrary fiel F, while in the remaining sections F is the fiel of real numbers R or of complex numbers C, which will be simultaneously enote by F We aopt stanar notations in the area: F[λ] enotes the ring of polynomials in the variable λ with coefficients in F an F(λ) enotes the fiel of fractions of F[λ], also known as the fiel of rational functions over F Vectors with entries in F[λ] are calle polynomial vectors In aition, F[λ] m n stans for the set of m n polynomial matrices, an F(λ) m n for the set of 2

3 m n rational matrices The egree of a polynomial vector, v(λ), or matrix, P (λ), is the highest egree of all its entries an is enote by eg(v) or eg(p ) The rigorous efinition of genericity we aopt requires to work insie the vector space over F of m n polynomial matrices with egree at most, which is enote by F[λ] m n Finally, F enotes the algebraic closure of F, I n the n n ientity matrix, an 0 m n the m n zero matrix, where the sizes will be omitte when they are clear from the context Polynomial matrices with size m n an egree at most are represente in this paper in the monomial basis as P (λ) = i=0 P iλ i, where P i F m n If P 0, then the egree of P (λ) is precisely, that is eg(p ) = The rank of P (λ) (sometimes calle normal rank ) is just the rank of P (λ) consiere as a matrix over the fiel F(λ), an is enote by rank(p ) The finite eigenvalues of P (λ) are the roots of its invariant polynomials, an associate to each such eigenvalue are elementary ivisors of P (λ) These an the rest of concepts use in this paper can be foun in the classical books [14, 15, 17], as well as in the summaries inclue in [6, Sect 2] an in [7, Sect 2], which are recommenable since are brief an follow exactly the conventions in this paper Since minimal basis is the key concept of this paper, we revise its efinition, characterization, main properties, an relate notions It is well known that every rational vector subspace V, ie, every subspace V F(λ) n over the fiel F(λ), has bases consisting entirely of polynomial vectors Among them some are minimal in the following sense introuce by Forney [13]: a minimal basis of V is a basis of V consisting of polynomial vectors whose sum of egrees is minimal among all bases of V consisting of polynomial vectors The funamental property [13, 17] of such bases is that the orere list of egrees of the polynomial vectors in any minimal basis of V is always the same Therefore, these egrees are an intrinsic property of the subspace V an are calle the minimal inices of V This iscussion immeiately leas us to the efinition of the minimal bases an inices of a polynomial matrix An m n polynomial matrix P (λ) with rank r smaller than m an/or n has non-trivial left an/or right rational null-spaces, respectively, over the fiel F(λ): N l (P ) := { y(λ) T F(λ) 1 m : y(λ) T P (λ) 0 T }, N r (P ) := { x(λ) F(λ) n 1 : P (λ)x(λ) 0 } Polynomial matrices with non-trivial left an/or right null-spaces are calle singular polynomial matrices If the rational subspace N l (P ) is non-trivial, it has minimal bases an minimal inices, which are calle the left minimal bases an inices of P (λ) Analogously, the right minimal bases an inices of P (λ) are those of N r (P ), whenever this subspace is non-trivial The efinition of minimal basis given above cannot be easily hanle in practice Therefore, [13, p 495] inclues five equivalent characterizations of minimal bases Among those, we present in Theorem 22 only the one we believe is the most useful in practice, since relies on the ranks of constant matrices The statement of Theorem 22 requires to introuce first Definition 21 For brevity, we use the expression column (resp, row) egrees of a polynomial matrix to mean the egrees of its column (resp, row) vectors Definition 21 Let 1,, n be the column egrees of N(λ) F[λ] m n The highest-columnegree coefficient matrix of N(λ), enote by N hc, is the m n constant matrix whose jth column is the vector coefficient of λ j in the jth column of N(λ) The polynomial matrix N(λ) is sai to be column reuce if N hc has full column rank Similarly, let 1,, m be the row egrees of M(λ) F[λ] m n The highest-row-egree coefficient matrix of M(λ), enote by M hr, is the m n constant matrix whose jth row is the vector coefficient of λ j in the jth row of M(λ) The polynomial matrix M(λ) is sai to be row reuce if M hr has full row rank Theorem 22 provies the announce characterization of minimal bases prove in [13] Theorem 22 The columns (resp, rows) of a polynomial matrix N(λ) over a fiel F are a minimal basis of the subspace they span if an only if N(λ 0 ) has full column (resp, row) rank for all λ 0 F, an N(λ) is column (resp, row) reuce 3

4 Remark 23 In this paper we follow the convention in [13] an often say, for brevity, that a p q polynomial matrix N(λ) is a minimal basis if the columns (when q < p) or rows (when p < q) of N(λ) form a minimal basis of the rational subspace they span Remark 24 If M(λ) F[λ] m k is a row reuce polynomial matrix, then M(λ) has full row (normal) rank This can be seen as follows: let i, i = 1,, m, be the row egrees of M(λ) an let S(λ) be an m m submatrix of M(λ) such that the corresponing submatrix of M hr, enote by S hr, is nonsingular Then, et S(λ) = λ 1+ + m et S hr + (lower egree terms in λ) 0 This means that the rows of M(λ) form a basis of the rational subspace V they span (which is the row space of M(λ)) an, by efinition of minimal basis, the sum of its row egrees m i=1 i is an upper boun for the sum of the minimal inices m i=1 η i of V Moreover, equality of the sums implies that M(λ) is a minimal basis of V an that the orere lists { i } an {η i } are also equal, by the uniqueness of the minimal inices Next, we introuce the concept of ual minimal bases, which has playe a key role in a number of recent applications [7, 9, 10, 18, 22] Definition 25 Polynomial matrices M(λ) F[λ] m k an N(λ) F[λ] n k with full row ranks are sai to be ual minimal bases if they are minimal bases satisfying m + n = k an M(λ) N(λ) T = 0 The name ual minimal bases seems to be very recent in the literature, because [7] is the first reference where is use, to our knowlege However, the origins of such concept can be foun in [13, Section 6] Accoring to [13, Section 6], ual minimal bases span rational vector subspaces of F(λ) k that are ual to each other In the language of null-spaces of matrix polynomials, observe that M(λ) is a minimal basis of N l (N(λ) T ) an that N(λ) T is a minimal basis of N r (M(λ)) As a consequence, the right minimal inices of M(λ) are the row egrees of N(λ) an the left minimal inices of N(λ) T are the row egrees of M(λ) Note that ual minimal bases have been efine with more columns than rows, as in the classical reference [13], although, one can also use matrices with more rows than columns in the efinition It follows from this iscussion that for every minimal basis, there exists a minimal basis that is ual to it In aition, every minimal basis is a minimal basis of some matrix polynomial The next theorem is crucial for the rest of this paper Its first part was proven in [13], while the secon (converse) part has been proven very recently in [7] Theorem 26 Let M(λ) F[λ] m (m+n) an N(λ) F[λ] n (m+n) be ual minimal bases with row egrees (η 1,, η m ) an (ε 1,, ε n ), respectively Then m η i = i=1 n ε j (21) j=1 Conversely, given any two lists of nonnegative integers (η 1,, η m ) an (ε 1,, ε n ) satisfying (21), there exists a pair of ual minimal bases M(λ) F[λ] m (m+n) an N(λ) F[λ] n (m+n) with precisely these row egrees, respectively 3 A finite number of rank conitions for minimal bases The characterization of a minimal basis given in Theorem 22 oes not seem tractable at first sight from a numerical point of view, since it requires a rank test over all λ 0 F In this section we show that this can be reuce to a finite number of rank tests on matrices that can be escribe from the imensions an coefficients of M(λ) A crucial role will be playe here by the so-calle Sylvester matrices [2, 4] of a polynomial matrix M(λ) of egree at most, M(λ) := M 0 + M 1 λ + + M λ, M i F m (m+n), (31) 4

5 efine for k = 1, 2, as M 0 M 1 M 0 M 1 S k := M M0, S k F (k+)m k(m+n), (32) 0 M M 1 } 0 0 {{ M } k blocks where S k has k block columns In orer to avoi trivialities, we assume throughout the paper that m > 0, n > 0, an > 0 In certain results involving Sylvester matrices of several polynomial matrices, we will use the notation S k (M) to inicate that the Sylvester matrices are associate to the polynomial matrix M(λ) The ranks of the matrices S k in (32) are funamental in this paper an the following two simple lemmas are use very often Lemma 31 Let S k for a given inex k > 1 have full column rank r k := k(m + n) Then all submatrices S l with 1 l < k also have full column rank r l := l(m + n) Proof Each matrix S l with 1 l < k, appropriately pae with zeros, forms the first l block columns of S k, from which the rank conition trivially follows Lemma 32 Let S k with a given inex k > 0 have full row rank r k := (k+)m Then all embeing matrices S l with k < l also have full row rank r l := (l + )m Proof We first prove that the result hols for l = k + 1 Since S k has full row rank r k, its bottom block row [ ] 0 M has full row rank m an so oes the matrix M Since the matrix S k+1 can be partitione in a block triangular form [ ] Sk X S k+1 =, 0 M where both S k an M have full row rank, S k+1 has also full row rank The result for general l larger than k, then easily follows by inuction In some of the results in this section, we will assume that M(λ) F[λ] m (m+n) has (full) normal rank m so that it has a (right) null-space of imension n (over the fiel of rational functions) The connection of this null-space N r (M) with the Sylvester matrices in (32) comes from the fact that any N(λ) F[λ] n (m+n) such that the columns of N(λ) T belong to N r (M) (ie M(λ)N(λ) T = 0) with expansion N(λ) := N 0 + N 1 λ + + N s λ s, N i F n (m+n), (33) will satisfy the equations S k N T 0 N T s 0 = 0 for all k s + 1 In particular, one can choose N(λ) T to be a minimal basis of N r (M) an then to erive certain rank conitions from this The following theorem is proven in [4] using this type of arguments 5

6 Theorem 33 Let M(λ) F[λ] m (m+n) be a polynomial matrix of full row rank m an let S k be its Sylvester matrices for k = 1, 2, Let r k be the rank of S k, n k be the right nullity of S k, an α k be the number of right minimal inices ε i of M(λ) equal to k Then these magnitues obey the following recursive relations α k = (n k+1 n k ) (n k n k 1 ) = (r k r k 1 ) (r k+1 r k ), k = 1, 2,, (34) initialize with r 0 = n 0 = 0 an α 0 = m + n r 1 = n 1 The secon equality in (34) between ranks an nullities easily follows from the ientity r k + n k = k(m + n), for k 1, (35) an an intuitive proof of the first ientity follows from the recursive efinition of the null-spaces of S k, for k = 1, 2, For k = 1 the theorem says that the matrix M 0 M 1 S 1 := F(+1)m (m+n) M has a right null space of imension n 1 equal to α 0, the number of right minimal inices ε i of M(λ) equal to 0 Inee, every inex ε i = 0 correspons to a right null vector of egree 0 (ie constant) of S 1 an hence also of M(λ) For k = 2, the matrix S 2 := M 0 0 M 1 M 0 M 0 M M 1 F (+2)m 2(m+n) has a right null space of imension n 2 equal to α 1 +2α 0, because S 2 contains S 1 (pae with zeros) twice as a submatrix an hence the null-space of S 1 will contribute also twice to the null-space of S 2 The aitional α 1 linearly inepenent vectors in the null-space of S 2 correspon to the true vectors of egree 1 in N r (M), ie, those vectors in the null-space of S 2 which cannot be obtaine as linear combinations of vectors in the null-space of S 1 pae with zeros It then follows that n 2 n 1 = α 0 + α 1 One uses the same arguments to show recursively that n k+1 n k = k α i, (36) i=0 which is equivalent to the first equality in (34) together with the initializations n 0 = 0, n 1 = α 0 We refer to [4] an the references therein for a more etaile proof The following corollary, also given in [4], is then easily erive from this Corollary 34 Let M(λ) F[λ] m (m+n) be a polynomial matrix of full row rank m an let S k be its Sylvester matrices for k = 1, 2, Let r k be the rank of S k an n k be the right nullity of S k If is the smallest inex k for which n k+1 n k = n, or equivalently r k+1 r k = m, (37) then is the maximum right minimal inex of M(λ) or, equivalently, the maximum column egree of any minimal basis of N r (M) Moreover, for all k larger than, the equalities (37) still hol 6

7 Proof As soon as the α i, the number of right minimal inices of M(λ) equal to i, a up to n, we have foun a complete polynomial basis of the rational right null-space of M(λ) There can be no further linearly inepenent right null vectors of M(λ) or nonzero α i since the sum k i=0 α i = n is the total number of right minimal inices of M(λ) The corresponing largest inex k with α k 0 is therefore the largest right minimal inex of M(λ) Since all α k = 0 for k larger than, equation (36) guarantees that the equality (37) continues to hol The equivalent conition r k+1 r k = m follows from (35) We can also express the sum of the right minimal inices ε j of M(λ) as a function of the nullities n k Corollary 35 Let M(λ) F[λ] m (m+n) be a polynomial matrix of full row rank m an let S k be its Sylvester matrices for k = 1, 2, Let n k an r k be the right nullity an the rank of S k, respectively, α k be the number of right minimal inices ε i of M(λ) equal to k, an be the maximum right minimal inex of M(λ) Then n ε j = kα k = k(n k 1 2n k + n k+1 ) = n n = r m (38) j=1 k=1 k=1 Proof The first ientity follows from the efinition of α k which is the number of right minimal inices ε j equal to k The secon ientity follows from Theorem 33 The thir ientity follows from the relations of Theorem 33, which written as follows n 0 = 0, α 0 α 1 α 2 α = n 1 n 2 n n +1, (39) allow us to see immeiately that k=1 kα k = n +1 ( + 1)n This equality combine with the ientity in Corollary 34, ie, n +1 = n + n, gives k=1 kα k = n n The fourth ientity follows from (35) applie to k = Remark 36 Notice that equation (39) in the proof of Corollary 35 establishes a one to one corresponence between the inex sets {n i, i = 1,, + 1} an {α i, i = 0,, }, as long as all the inices are non-negative an i=0 α i = n hols This last ientity is clearly equivalent to n +1 n = n We are now reay to formulate our necessary an sufficient finite number of rank conitions on constant matrices erive from M(λ) to verify that it is a minimal basis The conitions are given in the next theorem, where we note that m i=1 i, as a consequence of Corollary 34 an Theorem 26, an so the number of rank conitions to be checke is inee finite Theorem 37 Let M(λ) F[λ] m (m+n) be a polynomial matrix, let i, i = 1,, m, be its row egrees, an M hr be its highest-row-egree coefficient matrix Let S k be the Sylvester matrices of M(λ) for k = 1, 2,, an let r k an n k be the rank an the right nullity of S k, respectively Let be the smallest inex k for which n k+1 = n k + n, or equivalently, r k+1 = r k + m Then M(λ) is a minimal basis if an only if the following rank conitions are satisfie rank M hr = m an r m = m i (310) i=1 7

8 Proof Let us assume first that the conitions (310) hol The conition rank M hr = m says that M(λ) is row reuce, which implies that M(λ) has normal rank equal to its number of rows, m, (recall Remark 24) an that it has a right null space of imension n an a set of n right minimal inices ε j, j = 1,, n These right minimal inices can then be compute via the recurrences of Theorem 33 an conition r m = m i=1 i implies (accoring to Corollaries 35 an 34) that m i = i=1 n ε j (311) Since M(λ) has normal rank m, its rows form a polynomial basis of the row space of M(λ) whose egree sum equals m i=1 i If η 1,, η m are the minimal inices of the row space of M(λ), then Theorem 26 combine with (311) imply m i=1 i = n j=1 ε j = m j=1 η j, which in turn implies that the rows of M(λ) must form a minimal basis, by efinition of minimal basis The reverse implication follows immeiately: assume that M(λ) is a minimal basis Then rank M hr = m by Theorem 22 In aition, Corollaries 34 an 35 an Theorem 26 imply r m = m i=1 i Theorem 37 is consierably simplifie uner the aitional generic assumption that the leaing coefficient of M(λ) has full rank Corollaries 38 an 39 are two results in that irection Corollary 38 Let M(λ) = M 0 + M 1 λ + + M λ F[λ] m (m+n) be a polynomial matrix such that rank M = m Let S k be the Sylvester matrices of M(λ) for k = 1, 2,, let r k be the rank of S k, an let be the smallest inex k such that r k+1 = r k + m Then, M(λ) is a minimal basis if an only if r = m( + ) Proof Note that the assumptions of Corollary 38 imply that M hr = M an that the row egrees 1, 2,, m of M(λ) are all equal to an, so, m i=1 i = m In this scenario, if M(λ) is a minimal basis, then it satisfies the secon equality in (310), which implies r = m( + ) Conversely, if one assumes that r = m(+ ) hols, then the secon equality in (310) is satisfie, while the first one is guarantee by rank M = m Therefore M(λ) is a minimal basis Corollary 39 Let M(λ) = M 0 + M 1 λ + + M λ F[λ] m (m+n) be a polynomial matrix an let S k be the Sylvester matrices of M(λ) for k = 1, 2, Then, M(λ) is a minimal basis with rank M = m if an only if there exists an inex k such that S k has full row rank In this case, if is the smallest inex k for which S k has full row rank, then is the largest right minimal inex of M(λ) Proof If M(λ) is a minimal basis with rank M = m, then Corollary 38 implies r = m( + ), which is the number of rows of S Then S has full row rank Conversely, if there exists an inex k such that S k has full row rank, then M has full row rank, because the last block row of S k is [0 M ], which also implies that M(λ) has full row normal rank Let k 0 be the smallest inex k such that S k has full row rank an enote by r k the rank of any Sylvester matrix S k Then, accoring to Lemma 32, S k0 +1 also has full row rank an their ranks satisfy r k0 +1 r k0 = m (312) However, r k0 1 < (k )m, because S k0 1 has not full row rank Therefore, r k0 r k0 1 > m an, so, r k+1 r k > m for all k k 0 1, since Theorem 33 implies r j r j 1 r j+1 r j for all j 1 because α j 0 Therefore, k 0 is the smallest inex k such that r k+1 = r k + m, that is, k 0 = in Corollary 38 an r = m ( + ), since S k0 = S has full row rank So, Corollary 38 implies that M(λ) is a minimal basis In aition, observe that also k 0 = in Corollary 34 an, so, k 0 = is the largest right minimal inex of M(λ) j=1 8

9 We illustrate Theorems 33 an 37, an Corollaries 34, 38, an 39 with three simple examples With the purpose that the reaer can easily check that these results yiel the right outcome, in our three examples Theorem 22 also allows us to see very easily whether M(λ) is a minimal basis or not In aition, we isplay a matrix N(λ) such that the columns of N(λ) T are a minimal basis of N r (M) (an, so, N(λ) is a minimal basis ual to M(λ) whenever M(λ) is a minimal basis), which can again be easily checke by Theorem 22 Example 310 Let M(λ) F[λ] 6 8 an N(λ) F[λ] 2 8 be given: M(λ) = I 2 λi 2 I 2 λi 2 I 2 λi 2, N(λ) = [ λ 3 I 2 λ 2 I 2 λi 2 I 2 ] Then clearly M(λ)N(λ) T = 0 The ranks r k an nullities n k of the Sylvester matrices S k of M(λ) are: r 1 = 8, r 2 = 16, r 3 = 24, r 4 = 30; n 1 = 0, n 2 = 0, n 3 = 0, n 4 = 2 Then, clearly = 3 an (34) gives [α 0, α 1, α 2, α 3 ] = [0, 0, 0, 2] (which agrees with the row egrees of N(λ)) The conition (310) becomes rank M hr = 6 an r 3 3 m = = 6, which is inee equal to = 6 Therefore, M(λ) is a minimal basis with right minimal inices {3, 3} Observe that for proving just that M(λ) is a minimal basis (not for getting that its right minimal inices are {3, 3}), one can use simply Corollary 39 because the leaing coefficient of M(λ) has clearly full rank In this case the numbers of rows of S k for k = 1, 2, 3 are 12, 18, 24, respectively, which implies that S 3 has full row rank an that the largest right minimal inex of M(λ) is 3 The secon example correspons to a polynomial matrix that is not a minimal basis an has three ifferent right minimal inices Example 311 Let M(λ) F[λ] 4 7 an N(λ) F[λ] 3 7 be given: M(λ) = λ 0 1 λ 1 λ 1 λ, N(λ) = 0 1 λ 1 λ 2 λ 1 Then clearly M(λ)N(λ) T = 0 The ranks r k an nullities n k of the Sylvester matrices S k of M(λ) are: r 1 = 6, r 2 = 11, r 3 = 15; n 1 = 1, n 2 = 3, n 3 = 6 Then clearly = 2 an (34) gives [α 0, α 1, α 2 ] = [1, 1, 1] (which agrees with the row egrees of N(λ)) The conition (310) becomes rank M hr = 4 an r 2 2 m = = 3, which is not equal to = = 4 Therefore M(λ) is not a minimal basis Observe that for proving just that M(λ) is not a minimal basis (not for getting that its right minimal inices are {0, 1, 2}), one can use simply Corollary 38 because the leaing coefficient of M(λ) has clearly full rank In the previous two examples M(λ) has egree 1 an all its row egrees equal The polynomial matrix M(λ) in the next example oes not satisfy any of these two properties Example 312 Let M(λ) F[λ] 6 8 an N(λ) F[λ] 2 8 be given: M(λ) = I 2 λi 2 I 2 λi 2 I 2 λ 2 I 2, N(λ) = [ λ 4 I 2 λ 3 I 2 λ 2 I 2 I 2 ] 9

10 Then clearly M(λ)N(λ) T = 0 The ranks r k an nullities n k of the Sylvester matrices S k of M(λ) are: r 1 = 8, r 2 = 16, r 3 = 24, r 4 = 32, r 5 = 38; n 1 = 0, n 2 = 0, n 3 = 0, n 4 = 0, n 5 = 2 Then clearly = 4 an (34) gives [α 0, α 1, α 2, α 3, α 4 ] = [0, 0, 0, 0, 2] (which agrees with the row egrees of N(λ)) The conition (310) becomes rank M hr = 6 an r 4 4 m = = 8, which is equal to = 8 Therefore, M(λ) is a minimal basis In this case the leaing coefficient of M(λ) (the one corresponing to egree 2) has not full row rank an Corollaries 38 an 39 cannot be use We emphasize that in the examples above the rank conitions are completely in terms of the coefficient matrices of M(λ) via its Sylvester matrices, an that the minimal bases N(λ) are isplaye only for the purpose of comparison 4 Full-Sylvester-rank polynomial matrices an their properties In this section we characterize the polynomial matrices of size m (m + n) an egree at most all of whose Sylvester matrices S k efine in (32) have full rank, either full column rank when S k has more rows than columns or full row rank otherwise We avance that such matrices are always minimal bases an that satisfy other aitional properties The ceiling function of a real number x is often use in the rest of this paper an is enote by x Recall that x is the smallest integer that is larger than or equal to x The following efinition will allow us to refer to the property of interest in this section in a concise way Definition 41 Let M(λ) F[λ] m (m+n) be a polynomial matrix of egree at most, let S k for k = 1, 2, be its Sylvester matrices, an let r k be the rank of S k The polynomial matrix M(λ) is sai to have full-sylvester-rank if all the matrices S k have full rank, ie, if r k = min{(k+)m, k(m+n)} for k = 1, 2, It is necessary an sufficient to check at most two ranks for etermining whether a polynomial matrix has full-sylvester-rank or not, as a consequence of Lemmas 31 an 32 This is state in Lemma 42 Lemma 42 Let M(λ) F[λ] m (m+n) be a polynomial matrix of egree at most, let S k for k = 1, 2, be its Sylvester matrices, an let m k := an nk = m + t, where 0 t < n (41) n Then the following statements hol (a) k is the smallest inex k for which the number of columns of S k is larger than or equal to the number of rows of S k (b) If k > 1 an t > 0, then M(λ) has full-sylvester-rank if an only if S k 1 has full column rank an S k has full row rank (c) If k = 1 or t = 0, then M(λ) has full-sylvester-rank if an only if S k has full row rank Proof Part (a) follows from the size of S k isplaye in (32), because (k + )m k(m + n) is equivalent to m/n k, an this is equivalent to k k Part (b) follows from Lemmas 31 an 32, together with the fact that if t > 0 then S k has strictly more columns than rows, while S k 1 has strictly less columns than rows Note that S k 1 is efine since k > 1 In part (c) we have two scenarios If t = 0, then S k is a square matrix an the result follows again from Lemmas 31 an 32 for any value of k If k = 1 an t > 0, the same argument as in part (b) proves the result, with the only ifference that in this case S k 1 is not efine 10

11 Note that Corollary 39 immeiately implies that polynomial matrices with full-sylvester-rank are minimal bases whose leaing coefficient has full rank So, they have full row normal rank an, since the ranks r k of their Sylvester matrices are given by r k = min{(k + )m, k(m + n)}, their right minimal inices are fixe by the recurrence in Theorem 33 This leas to a characterization of full-sylvester-rank matrices in terms of their complete eigenstructure, ie, their finite an infinite elementary ivisors 1 an their left an right minimal inices This characterization together with other properties are presente in Theorem 43 Theorem 43 Let M(λ) = M 0 + M 1 λ + + M λ F[λ] m (m+n) be a polynomial matrix of egree at most, let α k be the number of right minimal inices of M(λ) equal to k, an let k an t be efine as in (41) Then the following statements hol (a) M(λ) has full-sylvester-rank if an only if the complete eigenstructure of M(λ) consists only of the following right minimal inices α k 1 = t, α k = n t, an α j = 0 for j / {k 1, k } (42) (b) If M(λ) has full-sylvester-rank, then M(λ) is a minimal basis with rank M = m, ie, with all its row egrees equal to (c) If M(λ) has full-sylvester-rank, then the egree of any minimal basis ual to M(λ) is equal to k That is, with the notation of Corollaries 34 an 35, = k hols for full-sylvester-rank matrices Proof Proof of (a) We assume first that M(λ) has full-sylvester-rank Then for all k k the Sylvester matrix S k of M(λ) has full row rank, since S k has full rank an oes not have more rows than columns Therefore, Corollary 39 implies that M(λ) is a minimal basis an that M has full row rank These properties imply in turn that the complete eigenstructure of M(λ) consists only of n right minimal inices, since M(λ) has not finite elementary ivisors as a consequence of Theorem 22, M(λ) has not infinite elementary ivisors as a consequence of rank M = m, M(λ) has not left minimal inices because has full row normal rank, an the number of right minimal inices of M(λ) is im N r (M) = m + n rank(m) = n It remains to etermine the n right minimal inices of M(λ) To this purpose note that the full-sylvester-rank property together with Lemma 42-(a) imply that S k has full column rank for all 1 k < k an has full row rank for all k k, which in terms of the rank, r k, an right nullity, n k, of S k is equivalent to r k = k(m + n), n k = 0, for 1 k < k, an r k = (k + )m, n k = nk m, for k k (43) The relations (43) can be expresse also in terms of k an t in (41) as follows r k = k(m + n), n k = 0, for 1 k < k, an r k = (k + )m, n k = n(k k ) + t, for k k (44) Finally, from Theorem 33 we have that α 0 = n 1 an α k = n k+1 2n k + n k 1 for k 1, which combine with (44) yiels α k = 0, for 1 k k 2, α k 1 = n k = t, α k = n k +1 2n k = n t, α k = 0, for k > k, (45) where the last line follows immeiately from the fact that the n right minimal inices of M(λ) have been alreay etermine in the previous lines of (45) Observe that accoring to (41) k 1 1 In this paper the infinite elementary ivisors of a polynomial matrix M(λ) with egree at most are the elementary ivisors associate to the eigenvalue 0 of the reversal polynomial matrix rev M(λ) := λ M(1/λ) [6] 11

12 an that in the limit case k = 1, ie, m n, we get from (45) α k 1 = α 0 = n 1 = t, which is consistent with the initialization in Theorem 33 In this limit case the first lines in (43), (44), an (45) are not present The following table illustrates the nullities, ranks, an α k numbers for k = k 2, k 1, k, k + 1, k + 2: k k 2 k 1 k k + 1 k + 2 n k 0 0 t t + n t + 2n r k (k 2)(m + n) (k 1)(m + n) (k + )m (k + + 1)m (k + + 2)m α k 0 t n t 0 0 Next we prove the sufficiency in part (a) Assume that the complete eigenstructure of M(λ) consists only of the n right minimal inices escribe in (42) Since the number of right minimal inices is precisely n, M(λ) has full normal rank, ie, rank(m) = m Then, we can apply Theorem 33 to M(λ) Note that (34) implies (36) an, so, the right nullities n k of the Sylvester matrices are etermine from the α k numbers as follows n 1 = α 0, n k+1 = n k + This recursion combine with (42) yiels k α i for k 1 (46) i=0 n k = 0, for k < k, n k = n(k k ) + t, for k k, which accoring to (44) implies that every Sylvester matrix S k of M(λ) has full rank This completes the proof of part (a) Part (b) has been alreay prove at the beginning of the proof of part (a) Part (c) follows from part (a) an the fact that the row egrees of any minimal basis ual to M(λ) are precisely the right minimal inices of M(λ) Theorem 43-(a) allows us to state in Theorem 44 another necessary an sufficient conition for a polynomial matrix to have full-sylvester-rank in terms only of its right minimal inices or, equivalently, the egrees of their ual minimal bases Theorem 44 Let M(λ) = M 0 + M 1 λ + + M λ F[λ] m (m+n) be a polynomial matrix of egree at most, let α k be the number of right minimal inices of M(λ) equal to k, an let k an t be efine as in (41) Then, M(λ) has full-sylvester-rank if an only if the right minimal inices of M(λ) are α k 1 = t, α k = n t, an α j = 0 for j / {k 1, k } (47) Proof Theorem 43-(a) implies that if M(λ) has full-sylvester-rank, then its right minimal inices are those in (47), which proves the necessity The sufficiency is prove as follows If the right minimal inices of M(λ) are those in (47), then im N r (M) = n an rank(m) = m Therefore, M(λ) has not left minimal inices In aition, the Inex Sum Theorem [6, Theorem 65] applie to the polynomial matrix M(λ) of grae implies that M(λ) has neither finite nor infinite elementary ivisors, since the sum of the right minimal inices of M(λ) is (k 1) α k 1 + k α k = (k 1) t + k (n t) = t + k n = m = rank(m) grae(m) Therefore, the complete eigenstructure of M(λ) consists only of the right minimal inices in (47) an Theorem 43-(a) implies that M(λ) has full-sylvester-rank Remark 45 Observe that Theorem 43-(b) states that full-sylvester-rank matrices are minimal bases with full rank leaing matrix coefficient M We emphasize that the converse result is not true: a minimal basis C(λ) F[λ] m (m+n) with egree at most an leaing coefficient C of full 12

13 rank has not necessarily full-sylvester-rank This follows immeiately from Theorem 26 because there exist ual minimal bases C(λ) F[λ] m (m+n) an D(λ) F[λ] n (m+n) with the row egrees of C(λ) all equal to (equivalently C has full rank) an with the row egrees of D(λ) having arbitrary values whose sum is m Since the row egrees of D(λ) are the right minimal inices of C(λ), they can be ifferent than those in (47) an so C(λ) has not full-sylvester-rank 5 Genericity of full-sylvester-rank matrices an consequences It is well-known that p q constant matrices have generically, ie, typically or almost always if the entries are consiere as ranom variables, full rank equal to min{p, q} Therefore, it is natural to expect that generically all of the Sylvester matrices of a polynomial matrix M(λ) F[λ] m (m+n) have full rank In other wors, it is natural to expect that generically a polynomial matrix M(λ) F[λ] m (m+n) has full-sylvester-rank However, this has to be rigorously prove, since Sylvester matrices are highly structure matrices containing many zero entries an with block columns intimately relate each other The evelopment of such rigorous proof an the analysis of some interesting consequences of this result are the goals of this section To this purpose, we nee to efine the precise meaning of genericity, which in this work is essentially the stanar notion in Algebraic Geometry We efine genericity insie the vector space F[λ] m (m+n) of polynomial matrices of size m (m + n) an egree at most, where in this section, an in the rest of the paper, F = R or F = C One motivation for consiering F[λ] m (m+n) as our ambient space comes from the applications to backwar error analyses of polynomial eigenproblems solve via linearizations or l-ifications (see [6, 9] for the efinition of l-ification) that we have in min for the results in this paper In practice, backwar errors are consiere arbitrary perturbations, since the only information available on them is their magnitue, which o not increase the egree of the polynomial matrix [10, 21] The first step in the efinition of genericity is to ientify F[λ] m (m+n) with R (+1)m(m+n) when F = R, or with R 2 (+1)m(m+n) when F = C If F = R such ientification can be mae, for instance, by mapping each polynomial matrix M(λ) = M 0 + M 1 λ + + M λ F[λ] m (m+n) into vec([m 0 M 1 M ]) R (+1)m(m+n), where vec( ) is the stanar vectorization operator efine for instance in [16, Chapter 4] If F = C, one consiers the entrywise real an imaginary parts of each matrix coefficient M i, enote by Re(M i ) an Im(M i ), respectively, an the ientification is mae by mapping M(λ) into vec([re(m 0 ) Im(M 0 ) Re(M ) Im(M )]) R 2(+1)m(m+n) Next, we recall that an algebraic set in R p is the set of common zeros of a finite number of multivariable polynomials with p variables an coefficients in R, an that an algebraic set is proper if it is not the whole set R p With these concepts at han, the stanar efinition of genericity of Algebraic Geometry is as follows: a generic set of R p is a subset of R p whose complement is containe in a proper algebraic set This efinition extens obviously to the corresponing one of generic set of F[λ] m (m+n), with F = R or F = C, through the bijective vec mappings iscusse above, since these mappings allow us to efine algebraic sets of F[λ] m (m+n) In the sequel, expressions as generically the polynomial matrices in F[λ] m (m+n) have the property P have the precise meaning of the polynomial matrices of F[λ] m (m+n) that satisfy property P are a generic set of F[λ] m (m+n) Now, we are in the position of stating an proving the main result of this section Theorem 51 Let Syl[λ] m (m+n) F[λ] m (m+n) be the set of polynomial matrices of size m (m + n), egree at most, an with full-sylvester-rank Then the complement of Syl[λ] m (m+n) a proper algebraic set of F[λ] m (m+n) an, so, Syl[λ] m (m+n) is a generic set of F[λ] m (m+n) Proof Let k an t be efine as in (41) We will prove the theorem in the case k > 1 an t > 0 The proofs in the cases k = 1 or t = 0 are similar an are omitte Taking into account Lemma is 13

14 42-(b), we have that Syl[λ] m (m+n) = {M(λ) F[λ] m (m+n) : et(sk 1 S k 1) et(s k Sk ) 0}, where S k 1 an S k are Sylvester matrices of M(λ) The complement of Syl[λ] m (m+n) is F[λ] m (m+n) ( Syl[λ] m (m+n) ) c m (m+n) = {M(λ) F[λ] : et(sk 1 S k 1) et(s k Sk ) = 0}, ( which is obviously an algebraic set More precisely: if F = R, Syl[λ] m (m+n) relative to ) c is the set of zeros of one multivariable polynomial in the entries of the matrix coefficients M i, i = 0, 1,,, of M(λ), an if F = C, the real an imaginary parts of et(sk 1 S k 1) et(s k Sk ) = 0 are equivalent to two multivariable polynomial equations in the real an imaginary parts of ( the entries of) the c matrix coefficients M i, i = 0, 1,,, of M(λ) It only remains to prove that is Syl[λ] m (m+n) proper, ie, that there is at least one polynomial matrix M(λ) F[λ] m (m+n) ( ) such that M(λ) / c This follows immeiately from Theorem 26, which guarantees the existence of Syl[λ] m (m+n) ual minimal bases M(λ) F[λ] m (m+n) an N(λ) F[λ] n (m+n) with all the row egrees of M(λ) equal to (so M(λ) F[λ] m (m+n) ), an with t row egrees of N(λ) equal to k 1 an the other n t equal to k, because the row egrees of each of these two matrices sum up m Therefore, the right minimal inices of M(λ), which are the row ( egrees on N(λ), ) are the ones in (47) an, by c Theorem 44, M(λ) Syl[λ] m (m+n) an M(λ) / Syl[λ] m (m+n) We have just prove that generically the polynomial matrices in F[λ] m (m+n) have full-sylvesterrank This can be combine with Theorem 43 to prove that generically the polynomial matrices in F[λ] m (m+n) satisfy other interesting properties, in particular, the property of being minimal bases whose egree- matrix coefficient has full row rank These properties are state in the following corollary, whose simple proof is omitte Corollary 52 Let k an t be efine as in (41) Then, the following subsets of F[λ] m (m+n) generic in F[λ] m (m+n) : (a) The set of m (m + n) polynomial matrices of egree at most an whose complete eigenstructure consists only of the following right minimal inices α k 1 = t, α k = n t, an α j = 0 for j / {k 1, k }, where α j enotes the number of right minimal inices equal to j (b) The set of m (m + n) polynomial matrices of egree at most that are minimal bases with egree- matrix coefficient of full row rank (c) The set of m (m + n) polynomial matrices of egree at most that are minimal bases an such that their ual minimal bases have egree equal to k Observe that the set efine in Corollary 52-(a) is precisely Syl[λ] m (m+n), while Syl[λ] m (m+n) is strictly inclue in the sets A an B efine in parts (b) an (c) of Corollary 52, respectively, as a consequence ( of the ) iscussion in Remark 45 Therefore the complements of these sets satisfy c ( ) c, A c an B c respectively Syl[λ] m (m+n) Syl[λ] m (m+n) are 14

15 6 Robustness of minimal bases an of full-sylvester-rank matrices This section has three goals: first, to characterize when a minimal basis M(λ) F[λ] m (m+n) is robust uner perturbations in the sense that all the polynomial matrices in a neighborhoo of M(λ) are also minimal bases; secon, to estimate the size of such neighborhoo; an, thir, to prove that full-sylvester-rank matrices are robust an to estimate the sizes of the corresponing neighborhoos of robustness Throughout the rest of the paper the singular values of a constant matrix A F p q are enote by σ 1 (A) σ 2 (A) σ min{p,q} (A) an the Sylvester matrices of any P (λ) F[λ] m (m+n) are enote by S k (P ) for k = 1, 2, In orer to stuy the questions of interest in this section, we efine a norm in F[λ] m (m+n) as follows: the norm of any P (λ) F[λ] m (m+n) is S 1 (P ) 2, where A 2 = σ 1 (A) is the stanar spectral norm of the matrix A [20] This norm inuces the istance S 1 (P ) S 1 ( P ) 2 = S 1 (P P ) 2 between any two polynomial matrices P (λ), P (λ) F[λ] m (m+n) In this section, as well as in the rest of this manuscript, we will use very often Lemma 62, whose proof relies on Lemma 61 Lemma 61 Let A = [ A 1 A 2 A k ] where Ai F m n i, i = 1,, k Then max σ 1 (A i ) σ 1 (A) i k i=1 σ1 2(A i) k max σ 1 (A i ) i Proof This is a simple consequence of [16, Corollary 313] an [16, Problem 22 in p 217] Lemma 62 Let P (λ) F[λ] m (m+n) Then the following inequalities hol for the Sylvester matrices of P (λ): S 1 (P ) 2 S k (P ) 2 k S 1 (P ) 2 Proof This is a irect consequence of Lemma 61 an the structure of S k (P ) Next theorem proves that a minimal basis of egree at most is robust insie F[λ] m (m+n) if an only if its row egrees are all equal to or, equivalently, if an only if its egree matrix coefficient has full rank Theorem 63 Let M(λ) = M 0 + M 1 λ + + M λ F[λ] m (m+n) be a minimal basis Then the following statements hol: (a) If rank M < m, then for all ɛ > 0 there exists a polynomial matrix M(λ) F[λ] m (m+n) that is not a minimal basis an satisfies S 1 (M) S 1 ( M) 2 < ɛ That is, as close as we want to M(λ) there are polynomial matrices that are not minimal bases (b) If rank M = m, then there exists an inex k such that S k (M) has full row rank an every polynomial matrix M(λ) = M 0 + M 1 λ + + M λ F[λ] m (m+n) that satisfies S 1 (M) S 1 ( M) 2 < σ (k+)m(s k (M)) k (61) is a minimal basis with rank M = m That is, all the polynomial matrices sufficiently close to M(λ) are minimal bases with full rank leaing coefficient Proof Proof of (a) Note that M has at least one zero row, because otherwise all the row egrees of M(λ) woul be equal to, then M woul be the highest-row-egree coefficient matrix of M(λ) (recall Definition 21), an rank M = m by Theorem 22, which is a contraiction Assume first m > 1, then either M 0 or M = 0 (in this latter case the egree of M(λ) is strictly less than ) In the case M 0, M has at least one zero row an at least one nonzero row 15

16 w k 0 A polynomial matrix M(λ) as in the statement can be constructe as follows: M(λ) is equal to M(λ) except that one of the zero rows of M is replace by the row vector (05ɛ/ w k 2 ) w k This implies that the highest-row-egree coefficient matrix M hr of M(λ) has two linearly epenent rows, so M(λ) is not a minimal basis by Theorem 22, an that S 1 (M) S 1 ( M) 2 = 05ɛ < ɛ In the case M = 0, a polynomial M(λ) as in the statement can be constructe as follows: M(λ) is equal to M(λ) except that two zero rows of M are both replace by the same arbitrary vector v ɛ with norm v ɛ 2 < ɛ/2 Then, M hr of M(λ) has two rows equal to v ɛ, so M(λ) is not a minimal bases by Theorem 22, an S 1 (M) S 1 ( M) 2 = [v ɛ ; v ɛ ] 2 < ɛ The limiting case m = 1, ie, when M(λ) has only one row an has egree smaller than, requires a ifferent proof Note that in this case rank M < m = 1 is equivalent to M = 0 Let us express M(λ) = [m 1 (λ) m 1+n (λ)] in terms of its entries From Theorem 22, we know that M(λ 0 ) 0 for any λ 0 F Given ɛ > 0, efine a number λ ɛ whose moulus is large enough to satisfy m i (λ ɛ )/λ ɛ < ɛ/ 1 + n for i = 1,, 1 + n Note that such number exists because all of the scalar polynomials m i (λ) have egree strictly smaller than In this case, we construct M(λ) as follows M(λ) = M(λ) λ [ m1 (λ ɛ ) λ ɛ m1+n(λ ɛ ) λ ɛ which satisfies M(λ ɛ ) = 0, so M(λ) is not a minimal basis, an S 1 (M) S 1 ( M) 2 < ɛ This completes the proof of part (a) Proof of (b) Corollary 39 guarantees the existence of a Sylvester matrix S k (M) with full row rank or, equivalently, with minimal singular value σ (k+)m (S k (M)) > 0 Conition (61) an Lemma 62 imply S k (M) S k ( M) 2 = S k (M M) 2 ], k S 1 (M M) 2 = k S 1 (M) S 1 ( M) 2 < σ (k+)m (S k (M)), which in turns implies that S k ( M) has full row rank by Weyl s perturbation theorem for singular values [20] Then, part (b) follows from applying Corollary 39 to M(λ) Remark 64 The natural choice of k in Theorem 63-(b) is the smallest inex k 0 for which S k (M) has full row rank, since in this way the enominator of the right han sie of (61) is the smallest possible one, which favors a larger estimation of the neighbourhoo of robustness However, note that the numerator plays a nontrivial role an another larger k might be a better choice As a consequence of the proof of Theorem 63-(a) when M = 0, it is obvious that minimal bases are never robust uner perturbations that increase their egrees This simple fact is state for completeness in the next corollary Corollary 65 Let M(λ) F[λ] p q be any minimal basis with p < q an let be any integer such that > eg(m) Then, there exist polynomial matrices in F[λ] p q which are not minimal bases an are as close as we want to M(λ) We have prove in Section 5 that generically the polynomial matrices in F[λ] m (m+n) have full- Sylvester-rank, which implies that generically the polynomials in F[λ] m (m+n) are minimal bases with all their row egrees equal to an with the row egrees of the minimal bases ual to them fully etermine by (47) It is not surprising, ue to their genericity, that full-sylvester-rank polynomial matrices are robust This is establishe in Theorem 66 together with estimates of the sizes of the robustness neighbourhoos We avance that Theorem 66 plays a key role in the perturbation theory of ual minimal bases evelope in Section 7, which is base on the mentione iea that the row egrees of the minimal bases ual to full-sylvester-rank matrices (or, equivalently, the right minimal inices of full-sylvester-rank matrices) are completely fixe (recall Theorems 43-(a) an 16