Matrix biorthogonal polynomials on the real line: Geronimus transformations

Gerardo Ariznabarreta∗,§, Juan C. Garćıa-Ardila†,¶, Manuel Mañas∗,‖ and Francisco Marcellán†,‡,∗∗ ∗Departamento de F́ısica Teórica II (Métodos Matemáticos de la F́ısica) Universidad Complutense de Madrid, Ciudad Universitaria Plaza de Ciencias 1, 28040 Madrid, Spain †Departamento de Matemáticas, Universidad Carlos III de Madrid Avd/Universidad 30, 28911 Leganés, Spain ‡Instituto de Ciencias Matemáticas (ICMAT) C/Nicolás Cabrera 13-15, 28049 Canto Blanco, Spain §gariznab@ucm.es ¶jugarcia@math.uc3m.es ‖manuel.manas@ucm.es ∗∗pacomarc@ing.uc3m.es


Introduction
Perturbations of a linear functional u in the linear space of polynomials with real coefficients have been extensively studied in the theory of orthogonal polynomials on the real line (scalar OPRL). In particular, when you deal with the positive definite case and linear functionals associated with probability measures supported in an infinite subset of the real line are considered, such perturbations provide interesting information in the framework of Gaussian quadrature rules taking into account the perturbation yields new nodes and Christoffel numbers, see [25,26]. Three perturbations have attracted the interest of the researchers. Christoffel perturbations, that appear when you consider a new functionalû = p(x)u, where p(x) is a polynomial, were studied in 1858 by the German mathematician Christoffel in [13], in the framework of Gaussian quadrature rules. He found explicit formulas relating the corresponding sequences of orthogonal polynomials with respect to two measures, the Lebesgue measure d µ supported in the interval (−1, 1) and dμ(x) = p(x)dµ(x), with p(x) = (x − q 1 ) · · · (x − q N ) a signed polynomial in the support of d µ, as well as the distribution of their zeros as nodes in such quadrature rules. Nowadays, these are called Christoffel formulas, and can be considered a classical result in the theory of orthogonal polynomials which can be found in a number of textbooks, see for example [12,26,71]. Explicit relations between the corresponding sequences of orthogonal polynomials have been extensively studied, see [25], as well as the connection between the corresponding monic Jacobi matrices in the framework of the so-called Darboux transformations based on the LU factorization of such matrices [9]. In the theory of orthogonal polynomials, connection formulas between two families of orthogonal polynomials allow to express any polynomial of a given degree n as a linear combination of all polynomials of degree less than or equal to n in the second family. A noteworthy fact regarding the Christoffel finding is that in some cases the number of terms does not grow with the degree n but remarkably, and on the contrary, remain constant, equal to the degree of the perturbing polynomial. See [25,26] for more on the Christoffel-type formulas. Christoffel transformations for orthogonal polynomials on the unit circle based on LU factorizations of CMV matrices have been studied in [10]. polynomials are greater than or equal to the degree of the perturbing polynomial. To end the section, we compare spectral versus nonspectral methods and present a number of applications. In particular, we deal with unimodular polynomial matrix perturbations and degree one matrix Geronimus transformations. Notice that in [6] we have extended these results to the matrix linear spectral case, i.e. to Uvarov-Geronimus-Christoffel formulas for certain matrix rational perturbations. Finally, an appendix with the definitions of Schur complements and quasideterminants is also included in order to have a perspective of these basic tools in the theory of matrix orthogonal polynomials.

On spectral theory of matrix polynomials
Here we give some background material regarding the spectral theory of matrix polynomials [39,52]. Definition 1. Let A 0 , A 1 , . . . , A N ∈ C p×p be square matrices of size p × p with complex entries and A N = 0 p . Then is said to be a matrix polynomial of degree N , deg(W (x)) = N . The matrix polynomial is said to be monic when A N = I p , where I p ∈ C p×p denotes the identity matrix. The linear space -a bimodule for the ring of matrices C p×p -of matrix polynomials with coefficients in C p×p will be denoted by C p×p [x].  (2) Definition 6 (Canonical Jordan pairs). We also define the corresponding canonical Jordan pair (X a , J a ) with X a the matrix is nonsingular.

Definition 7 (Jordan triple). Given
with Y a ∈ C αa×p , we say that (X, J, Y ) is a Jordan triple whenever  Moreover, [39,Theorem 1.23], gives the following characterization.
Here, given a function f (x) we use the following notation for its derivatives evaluated at an eigenvalue x a ∈ σ(W (x)) In this paper, we assume that the partial multiplicities are ordered in an increasing way, i.e. κ  (4)

Definition 8 (Spectral jets).
Given a matrix function f (x) smooth in region Ω ⊂ C with x a ∈ Ω, a point in the closure of Ω we consider its matrix spectral jets and given a Jordan pair the root spectral jet vectors i (x a ), . . . , n;1 , . . . , Q (a) n;sa ∈ C p×αa ,

Lemma 1 (Root spectral jets and Jordan pairs).
Given a canonical Jordan pair (X, J) for the monic matrix polynomial W (x) we have that Thus, any polynomial P n (x) = n j=0 P j x j has as its spectral jet vector corresponding to W (x) the following matrix. J P = P 0 X + P 1 XJ + · · · + P n XJ n−1 .
Lemma 2. Given a Jordan triple (X, J, Y ) for the monic matrix polynomial W (x) we have which is nonsingular, see Propositions 8 and 9. The biorthogonality condition (2.6) of [39] for R and Q is and if (X, J, Y ) is a canonical Jordan triple, then Regarding the matrix B, we have the following.
Definition 11. Let us consider the bivariate matrix polynomial where A j are the matrix coefficients of W (x), see (1).
We consider the complete homogeneous symmetric polynomials in two variables x j y n−j .
For example, the first four polynomials are

Proposition 13.
In terms of complete homogeneous symmetric polynomials in two variables we can write

On orthogonal matrix polynomials
The where φ k ∈ C p×p . Thus, we can identify the dual of the right module with the corresponding left submodule. This dual is a free module with a unique rank, equal to m + 1, and a dual basis We have similar statements for the left module C p×p m [x], being its dual a right module Definition 12 (Sesquilinear form). A sesquilinear form ·, · on the bimodule C p×p [x] is a continuous map such that for any triple P (x), Q(x), R(x) ∈ C p×p [x] the following properties are fulfilled: The reader probably has noticed that, despite dealing with complex polynomials in a real variable, we have followed [26] and chosen the transpose instead of the Hermitian conjugated. For any couple of matrix polynomials P (x) = deg P k=0 p k x k and Q(x) = deg Q l=0 q l x l the sesquilinear form is defined by where the coefficients are the values of the sesquilinear form on the basis of the module The corresponding semi-infinite matrix i.e. a p × p matrix of Borel measures supported in R. Given any pair of matrix we introduce the following sesquilinear form: A more general sesquilinear form can be constructed in terms of generalized functions (or continuous linear functionals). In [53,54], a linear functional setting for orthogonal polynomials is given. We consider the space of polynomials C[x], with an appropriate topology, as the space of fundamental functions, in the sense of [32,33], and take the space of generalized functions as the corresponding continuous linear functionals. It is remarkable that the topological dual space coincides with the algebraic dual space. On the other hand, this space of generalized functions is the space of formal series with complex coefficients ( In this paper, we use generalized functions with a well-defined support and, consequently, the previously described setting requires a suitable modification. Following [32,33,67], let us recall that the space of distributions is a space of generalized functions when the space of fundamental functions is constituted by the complex-valued smooth functions of compact support D := C ∞ 0 (R), the so-called space of test functions. In this context, the set of zeros of a distribution u ∈ D is the region Ω ⊂ R if for any fundamental function f (x) with support in Ω we have u, f = 0. Its complement, a closed set, is what is called support, supp u, of the distribution u. Distributions of compact support, u ∈ E , are generalized functions for which the space of fundamental functions is the topological space of complex-valued The set of distributions of compact support is a first example of an appropriate framework for the consideration of polynomials and supports simultaneously. More general settings appear within the space of tempered distributions S , S D . The space of fundamental functions is given by the Schwartz space S of complex-valued fast decreasing functions, see [32,33,67]. We consider the space of fundamental functions constituted by smooth functions of slow growth O M ⊂ E, whose elements are smooth functions with derivatives bounded by polynomials. D . Then, we consider the space of fast decreasing distributions O c given by those distributions u ∈ D such that for each positive integer k, we have that ( , with deg P = k, can be writ- Therefore, given a fast decreasing distribution u ∈ O c we may consider Moreover, it can be proven that O M O c , see [53]. Summarizing this discussion, we have found three generalized function spaces suitable for the discussion of polynomials and supports simultaneously: The linear functionals could have discrete and, as the corresponding Gram matrix is required to be quasi-definite, infinite support. Then, we are faced with discrete orthogonal polynomials, see for example [57]. Two classical examples are those of Charlier and Meixner. For µ > 0 we have the Charlier (or Poisson-Charlier) linear functional and β > 0 and 0 < c < 1, the Meixner linear functional is See [4] for matrix extensions of these discrete linear functionals and corresponding matrix orthogonal polynomials.

Definition 13 (Hankel sesquilinear forms). Given a matrix of generalized functions as entries
i.e. u i,j ∈ (C[x]) , then the associated sesquilinear form P (x), Q(x) u is given by When u k,l ∈ O c , we write u ∈ (O c ) p×p and say that we have a matrix of fast decreasing distributions. In this case the support is defined as supp(u) := N k,l=1 supp(u k,l ). G. Ariznabarreta et al. Observe that in this Hankel case, we could also have continuous and discrete orthogonality.

Proposition 14. In terms of the moments
the Gram matrix of the sesquilinear form given in Definition 13 is the following moment matrix : of Hankel-type.

Matrices of generalized kernels and sesquilinear forms
The previous examples all have in common the same Hankel block symmetry for the corresponding matrices. However, there are sesquilinear forms which do not have this particular Hankel-type symmetry. Let us stop for a moment at this point, and elaborate on bilinear and sesquilinear forms for polynomials. We first recall some facts regarding the scalar case with p = 1, and bilinear forms instead of sesquilinear forms. Given u x,y ∈ (C[x, y]) = (C[x, y]) * ∼ = C[[x, y]], we can consider the continuous bilinear form B(P (x), Q(y)) = u x,y , P (x) ⊗ Q(y) . This gives a continuous linear map L u : C[y] → (C[x]) such that B(P (x), Q(y)) = L u (Q(y)), P (x) . The Gram matrix of this bilinear form has coefficients G k,l = B(x k , y l ) = u x,y , x k ⊗ y l = L u (y l ), x k . Here we follow Schwartz discussion on kernels and distributions [66], see also [47]. A kernel u(x, y) is a complex-valued locally integrable function, that defines an integral operator f (x) → g(x) = u(x, y)f (y)d y. Following [67] we denote (D) x and (D ) x the test functions and the corresponding distributions in the variable x, and similarly for the variable y. We extend this construction considering a bivariate distribution in the variables x, y, u x,y ∈ (D ) x,y , that Schwartz called noyau-distribution, and as we use a wider range of generalized functions we will call generalized kernel. This u x,y generates a continuous bilinear form B u φ(x), ψ(y) = u x,y , φ(x) ⊗ ψ(y) . It also generates a continuous linear map L u :  [66], the generalized kernel u x,y is such that L u : (E) y → (E ) x if and only if the support of u x,y in R 2 is compact. a We can extended these ideas to the matrix scenario of this paper, where instead of bilinear forms we have sesquilinear forms.

Definition 14.
Given a matrix of generalized kernels x,y provides a continuous sesquilinear form with entries given by where L u k,l : C[y] → (C[x]) -or depending on the setting L u k,l : ) a continuous linear map. Or, in other scenarios If, instead of a matrix of bivariate distributions, we have a matrix of bivariate measures then we could write for the sesquilinear form P (x), Q(y) = P (x)d µ(x, y)(Q(y)) , where µ(x, y) is a matrix of bivariate measures. For the scalar case p = 1, Adler and van Moerbeke discussed in [1] different possibilities of non-Hankel Gram matrices. Their Gram matrix has as coefficients G k,l = u l , x k , for an infinite sequence of generalized functions u l , that recovers the Hankel scenario for u l = x l u. They studied in more detail the following cases: (ii) Concatenated solitons: We see that the three last weights are generalized functions. To compare with the Schwartz's approach we observe that u x,y , x k ⊗ y l = u l , x k and, consequently, we deduce u l = L u (y l ) (and for continuous kernels u l (x) = u(x, y)y l d y). The first case has a banded structure and its Gram matrix fulfills Λ m G = G(Λ ) m . In [3], different examples are discussed for the matrix orthogonal polynomials, like bigraded Hankel matrices Λ n G = G Λ m , where n, m are positive integers, can be realized as G k,l = u l , I p x k , in terms of matrices of linear functionals u l which satisfy the following periodicity condition u l+m = u l x n . Therefore, given the linear functionals u 0 , . . . , u m−1 we can recover all the others.

Sesquilinear forms supported by the diagonal and Sobolev sesquilinear forms
First we consider the scalar case

Proposition 15 (Sobolev bilinear forms). The bilinear form corresponding to a generalized kernel supported by the diagonal is
For order zero u (n,m) x generalized functions, i.e. for a set of Borel measures µ (n,m) , we have which is of Sobolev type. Thus, in the scalar case, generalized kernels supported by the diagonal are just Sobolev bilinear forms. The extension of these ideas to the matrix case is immediate, we only need to require to all generalized kernels to be supported by the diagonal.

Geronimus transformations for matrix biorthogonal polynomials
For a recent review on scalar Sobolev orthogonal polynomials see [51]. Observe that with this general framework we could consider matrix discrete Sobolev orthogonal polynomials, that will appear whenever the linear functionals u (m,n) have infinite discrete support, as far as u is quasi-definite.

Definition 17 (Quasi-definiteness). A Gram matrix of a sesquilinear form
We say that the bivariate generalized function u x,y is quasi-definite and the corresponding sesquilinear form is nondegenerate whenever its Gram matrix is quasidefinite.
Proposition 17 (Gauss-Borel factorization, see [7]). If the Gram matrix of a sesquilinear form ·, · u is quasi-definite, then there exists a unique Gauss-Borel factorization given by where S 1 , S 2 are lower unitriangular block matrices and H is a diagonal block matrix 1950007-17 Bull. Math. Sci. 2019.09. Downloaded from www.worldscientific.com by UNIVERSIDAD COMPLUTENSE MADRID LIBRARY on 10/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

G. Ariznabarreta et al.
For l ≥ k we will also use the following bordered truncated Gram matrix: where we have replaced the last row of blocks of the truncated Gram matrix G [k] by the row of blocks [G l,0 , . . . , G l,k−1 ]. We also need a similar matrix but replacing the last block column of G [k] by a column of blocks as indicated Using last quasideterminants, see [27,58] and Appendix A, we find the following.
and for the inverse elements [58] the formulas We see that the matrices H k are quasideterminants, and following [7,8] we refer to them as quasi-tau matrices.

Biorthogonal polynomials, second kind functions and
Christoffel-Darboux kernels

Remark 1.
Observe that the Gram matrix can be expressed as and its block entries are If the sesquilinear form derives from a matrix of bivariate measures µ(x, y) = [µ i.j (x, y)] we have for the Gram matrix blocks which reduces for absolutely continuous measures with respect to the Lebesgue measure d xd y to a matrix of weights w(x, y) = [w i,j (x, y)], and When the matrix of generalized kernels is Hankel we recover the classical Hankel structure, and the Gram matrix is a moment matrix. For example, for a matrix of measures we will have G k,l = x k+l d µ(x).

Definition 19.
Given a quasi-definite matrix of generalized kernels u x,y and the Gauss-Borel factorization (17) respectively.

Proposition 19 (Biorthogonality).
Given a quasi-definite matrix of generalized kernels u x,y , the first and second families of monic matrix polynomials P [1] n (x) ∞ n=0 and P [2] n (x) ∞ n=0 are biorthogonal Remark 2. The biorthogonal relations yield the orthogonality relations

Remark 3 (Symmetric generalized kernels).
If u x,y = (u y,x ) , the Gram matrix is symmetric G = G and we are dealing with a Cholesky block factorization with S 1 = S 2 and H = H . Now P [1] n (x) = P [2] n (x) =: P n (x), and {P n (x)} ∞ n=0 is a set of monic orthogonal matrix polynomials. In this case C [1] n (x) = C [2] n (x) =: C n (x).
The shift matrix is the following semi-infinite block matrix: which satisfies the spectral property

Proposition 20. The symmetry of the block Hankel moment matrix reads
Notice that this symmetry completely characterizes Hankel block matrices.

Geronimus transformations for matrix biorthogonal polynomials
The reader must notice the abuse in the notation. But for the sake of simplicity we have used the same letter for Jacobi and Jordan matrices. The type of matrix will be clear from the context. [2] (x). and the second kind functionsá la Gram satisfy

Proposition 22.
For Hankel-type Gram matrices (i.e. associated with a matrix of univariate generalized functionals) the two Jacobi matrices are related by H −1 J 1 = J 2 H −1 , being, therefore, a tridiagonal matrix. This yields the three-term relation for biorthogonal polynomials and second kind functions, respectively.

Proposition 23.
We have the following last quasideterminantal expressions: the sesquilinear form ·, · u , we define the nth Christoffel-Darboux kernel matrix polynomial and the mixed Christoffel-Darboux kernel k (x).

Proposition 24.
(i) For a quasi-definite matrix of generalized kernels u x,y , the corresponding Christoffel-Darboux kernel gives the projection operator (ii) In particular, we have

and the mixed Christoffel-Darboux kernel fulfills
Proof. We only prove the second formula, for the first one proceeds similarly. It is obviously a consequence of the three-term relation. First, let us notice that [2] (y)) J 2 H −1 = y(P [2] (y)) H −1 .
Using this, we calculate the (P [2] [n] (y)) [ [n] (x), first by computing the action of middle matrix on its left and then on its right to get n−1 (x), and since P 0 = I p the proposition is proven.
Next, we deal with the fact that our definition of second kind functions implies non-admissible products and do involve series.

Definition 22.
For the support of the matrix of generalized kernels supp(u x,y ) ⊂ C 2 we consider the action of the component projections π 1 , π 2 : C 2 → C on its first and second variables, (x, y) → y, respectively, and introduce the projected supports supp x (u) := π 1 (supp(u x,y )) and supp y (u) := π 2 (supp(u x,y )), both subsets of C. We will assume that r x := sup{|z| : z ∈ supp x u} < ∞ and r y := sup{|z| : z ∈ supp y u} < ∞ We also consider the disks about infinity, or annulus around the origin, D x := {z ∈ C : |z| > r x } and D y := {z ∈ C : |z| > r y }.

Definition 23 (Second kind functionsá la Cauchy).
For a generalized kernel it is such that u x,y ∈ ((O c ) x,y ) p×p we define two families of second kind functionś a la Cauchy given by C [1] n (z) = P [1] n (x),

Matrix Geronimus Transformations
Geronimus transformations for scalar orthogonal polynomials were first discussed in [35], where some determinantal formulas were found, see [55,73] G. Ariznabarreta et al. in [17] and in the general case in [16]. Here we discuss its matrix extension for general sesquilinear forms.

Definition 24. Given a matrix of generalized kernels
with a given support supp u x,y , and a matrix polynomial W (y) ∈ C p×p [y] of degree N , such that σ(W (y)) ∩ supp y (u) = ∅, a matrix of bivariate generalized functionsǔ x,y is said to be a matrix Geronimus transformation of the matrix of generalized kernels u x,y if

Proposition 26. In terms of sesquilinear forms a Geronimus transformation fulfills
while, in terms of the corresponding Gram matrices, satisfieš We will assume that the perturbed moment matrix has a Gauss-Borel factor- Hence, the Geronimus transformation provides the family of matrix biorthogonal polynomialsP with respect to the perturbed sesquilinear form ·, · ǔ . Observe that the matrix generalized kernels v x,y such that v x,y W (y) = 0 p can be added to a Geronimus transformed matrix of generalized kernelsǔ x,y →ǔ x,y + v x,y , to get a new Geronimus transformed matrix of generalized kernels. We call masses these type of terms.

The resolvent and connection formulas
Definition 25. The resolvent matrix is The key role of this resolvent matrix is determined by the following properties.
where the products in the right-hand side are associative.
(ii) The resolvent matrix is a lower unitriangular block banded matrix -with only the first N block subdiagonals possibly not zero, i.e.
(iv) For the last subdiagonal of the resolvent we have Proof. (i) From Proposition 26 and the Gauss-Borel factorization of G andǦ we get (ii) The resolvent matrix, being a product of lower unitriangular matrices, is a lower unitriangular matrix. However, from (17) we deduce that it is a matrix with all its subdiagonals with zero coefficients but for the first N . Thus, it must have the described band structure. (iii) From the definition we have (18). Let us notice that (17) can be written as and (19) follows. (iv) It is a consequence of (17). The connection formulas (18) and (19) can be written aš P [1] n (x) = P [1] n (x) + with B given in Definition 10.

Geronimus transformations for matrix biorthogonal polynomials
and using the Gauss-Borel factorization the result follows. For (25) we have (Č [2] Observe that the corresponding entries are
Proof. For the first connection formula (27) we consider the pairing and compute it in two different ways. From (21) we get and, therefore, K n−1 (x, y) =Ǩ n−1 (x, y). Relation (22) leads to and (27)  which, as before, can be computed in two different forms. On the one hand, using (24) we get N ] is the truncation to the first n block rows and first N block columns ofȞ Š 2 − . This simplifies for n ≥ N to On the other hand, from (22) we conclude and, consequently, we obtaiň . . .

Spectral jets and relations for the perturbed polynomials and its second kind functions
For the time being we will assume that the perturbing polynomial is monic, Definition 27. Given a perturbing monic matrix polynomial W (y) the most general mass term will have the form with ξ [a] j,m ∈ C p .
so that, with the particular choice in (29), we get the diagonal case.

Definition 30. The left Jordan chain matrix is given by
For z = x a , we also introduce the p × p matriceš C (a) n;i (z) := P [1] n (x), (ξ where i = 1, . . . , s a .

Remark 6.
Assume that the mass matrix is as in (30). Then, in terms of we can write Consequently,Č (a) Observe that X is a block upper triangular matrix, with blocks in C p×p .
Proof. Notice that we can write where the C p -valued function T (a,b) (x) is analytic at x = x b and, in particular, Proof. First, for the functionČ (a) where the C p -valued function T (a,b) (x) is analytic at x = x b . Second, from (31) and Lemma 4 we deduce thať j (x) = P [1] n (x), (ξ i T (a,a) (x), and the result follows.
We evaluate now the spectral jets of the second kind functionsČ [1] (z)á la Cauchy, thus we must take limits of derivatives precisely in points of the spectrum of W (x), which do not lay in the region of definition but on the border of it. Notice that these operations are not available for the second kind functionś a la Gram.

Now, using (36) we can write the second equation as
A similar argument leads to the second relation in (39).

Definition 32.
For the Hankel masses, we also consider the matrices T

Remark 7.
In the next results, the jets of the Christoffel-Darboux kernels are considered with respect to the first variable x, and we treat the y-variable as a parameter. (29), we have the following last quasideterminantal expressions for the perturbed biorthogonal matrix polynomials and its matrix norms: [1] n−N − P [1] n−N (x), (ξ) x W P [1] n−N (x) . . . . . . [1] n−N − P [1] n−N (x), (ξ) x W H n−N J C [1] n−N +1
and the result follows. To get the transformation for the H's we proceed as follows.

Geronimus transformations for matrix biorthogonal polynomials
Therefore, the corresponding spectral jets do satisfy . .
Now, for n ≥ N , from Definition 26 and the fact that ω n,n−N =Ȟ n H n−N −1 , we get , and the result follows.

Nonspectral Christoffel-Geronimus formulas
We now present an alternative orthogonality relations approach for the derivation of Christoffel-type formulas, that avoids the use of the second kind functions and of the spectral structure of the perturbing polynomial. A key feature of these results is that they hold even for perturbing matrix polynomials with singular leading coefficient.
Definition 33. For a given perturbed matrix of generalized kernelsǔ x,y = u x,y (W (y)) −1 + v x,y , with v x,y W (y) = 0 p , we define a semi-infinite block matrix Remark 8. Its blocks are R n,l = P [1] n (x), I p y l ǔ ∈ C p×p . Observe that for a Geronimus perturbation of a Borel measure d µ(x, y), with general masses as in (29) we have R n,l = P [1] n (x)d µ(x, y)(W (y)) −1 y l 1 m! P [1] n (x), ξ Geronimus transformations for matrix biorthogonal polynomials that, when the masses are discrete and supported by the diagonal y = x, reduces to R n,l = P [1] n (x)d µ(x, y)(W (y) Proposition 32. The following relations hold true: Proof. (44) follows from Definition 33. Indeed, To deduce (45) we recall (16), (44), and the Gauss factorization of the perturbed matrix of moments Finally, to get (46), we use (17) together with (45), which implies ω = ωRW (Λ ) S 2 ) H −1 , and as the resolvent it is unitriangular with a unique inverse matrix [14], we obtain the result.
we construct a submatrix of it by selecting N p columns among all the np columns.
For that aim, we use indexes (i, a) labeling the columns, where i runs through The set of indexes I is said to be poised if R n is nonsingular. We also use the notation where r n := [c (i1,a1) , . . . ,c (iNp,aNp) ]. Herec (ir ,ar) denotes the a r th column of the matrix R n,ir . Given a poised set of indexes we define (r K n (y)) as the matrix built up by taking from the matrices r K n,ir (y) the columns a r .

Geronimus transformations for matrix biorthogonal polynomials
As the truncation R [n] is nonsingular, this matrix is full rank, i.e. all its N p rows are linearly independent. Thus, there must be N p independent columns and the desired result follows. Lemma 8. Whenever the leading coefficient A N of the perturbing polynomial W (y) is nonsingular, we can decompose any monomial I p y l as We will show now that this is the unique solution to this linear system. Let us proceed by contradiction and assume that there is another solution, say [ω n,n−N , . . . ,ω n,n−1 ]. Consider then the monic matrix polynomial P n (x) = P [1] n (x) +ω n,n−1 P [1] n−1 (x) + · · · +ω n,n−N P [1] n−N (x).
1950007-45 Bull. Math. Sci. 2019.09. Downloaded from www.worldscientific.com by UNIVERSIDAD COMPLUTENSE MADRID LIBRARY on 10/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

Spectral versus nonspectral
Definition 36. We introduce the truncation given by taking only the first N columns of a given semi-infinite matrix Then, we can connect the spectral methods and the nonspectral techniques as follows Proposition 38. The following relation takes place Proof. From (24) we deduce thať Taking the corresponding root spectral jets, we obtain [1] , that, together with (39), gives Now, relation (45) implies But, given that ω is a lower unitriangular matrix, and therefore with an inverse, see [14], the unique solution to ωX = 0, where X is a semi-infinite matrix, is X = 0. We now discuss an important fact, which ensures that the spectral Christoffel-Geronimus formulas presented in previous sections make sense. [1] n−N − P [1] n−N (x), (ξ) x W . . .
Now, Proposition 36 and Lemma 2 lead to the result.
We stress at this point that (48) connects the spectral and the nonspectral methods. Moreover, when we border with a further block row we obtain      J C [1] n−N − P [1] n−N (x), (ξ) x W . . .

Unimodular Christoffel perturbations and nonspectral techniques
The spectral methods apply to those Geronimus transformations with a perturbing polynomial W (y) having a nonsingular leading coefficient A N . This was also the case for the techniques developed in [2] for matrix Christoffel transformations, where the perturbing polynomial had a nonsingular leading coefficient. However, we have shown that despite we can extend the use of the spectral techniques to the study of matrix Geronimus transformations, we also have a nonspectral approach applicable even for singular leading coefficients. For example, some cases that have appeared several times in the literature -see [21] -are unimodular perturbations and, consequently, with W (y) having a singular leading coefficient. In this case, we have that (W (y)) −1 is a matrix polynomial, and we can consider the Geronimus transformation associated with the matrix polynomial (W (y)) −1 -as the spectrum is empty σ(W (y)) = ∅, no masses appear -as a Christoffel transformation with perturbing matrix polynomial W (y) of the original matrix of generalized kernelš u x,y = u x,y (W (y)) −1 −1 = u x,y W (y). (49) We can apply Theorem 3 with R = P [1] (x), χ(y) uW , R n,l = P [1] n (x), I p y l uW ∈ C p×p .
For example, when the matrix of generalized kernels is a matrix of measures µ, we can write R n,l = P [1] n (x)d µ(x, y)W (y)y l .
Here W (x) is a Christoffel perturbation and deg((W (x)) −1 ) gives you the number of original orthogonal polynomials required for the Christoffel-type formula. Theorem 3 can be nicely applied to getP [1] n (x) andȞ n . However, it only gives Christoffel-Geronimus formulas for (P [2] n (y)) A N and given that A N is singular, we only partially recoverP [2] n (y). This problem disappears whenever we have symmetric generalized kernels u x,y = (u y,x ) , see Remark 3, as then P [1] n (x) = P [2] n (x) =: P n (x) and biorthogonality collapses to orthogonality of {P n (x)} ∞ n=0 . From (49), we need to require u x,y W (y) = (W (x)) (u y,x ) , that when the initial matrix of kernels is itself symmetric u x,y = (u y,x ) reads u x,y W (y) = (W (x)) u x,y . Now, if we are dealing with Hankel matrices of generalized kernels u x,y = u x,x we find u x,x, W (x) = (W (x)) u x,x , that for the scalar case reads u x,x = u 0 I p with u 0 a generalized function we need W (x) to be a symmetric matrix polynomial. For this scenario, if {p n (x)} ∞ n=0 denotes the set of monic orthogonal polynomials associated with u 0 , we have R n,l = u 0 , p n (x)W (x)x l .
For example, if we take p = 2, with the unimodular perturbation given by we have, that the inverse is the following matrix polynomial: where det W (x) is a constant, and the inverse has also degree 2. Therefore, for n ∈ {2, 3, . . .}, we have the following expressions for the perturbed matrix orthogonal 1950007-50 Bull. Math. Sci. 2019.09. Downloaded from www.worldscientific.com by UNIVERSIDAD COMPLUTENSE MADRID LIBRARY on 10/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. and the inverse of a general unimodular matrix polynomial can be computed immediately once its factorization in terms of elementary matrices is given. However, the degree of the matrix polynomial and its inverse requires a separate analysis.

Appendix A. Schur Complements and Quasideterminants
We first notice that the Schur complement was not introduced by Schur but by Haynsworth in 1968 in [44,45]. Haynsworth coined that named because the Schur determinant formula given in what today is known as Schur lemma in [65]. For an ample overview on Schur complement and many of its applications see [72]. The most easy examples of quasideterminants are Schur complements. Gel'fand and collaborators have made many essential contributions to the subject, see [27] for an excellent survey on the subject. Olver's on a paper on multivariate interpola-1950007-60 Bull. Math. Sci. 2019.09. Downloaded from www.worldscientific.com by UNIVERSIDAD COMPLUTENSE MADRID LIBRARY on 10/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles.
Geronimus transformations for matrix biorthogonal polynomials tion, see [58], discusses an alternative interesting approach to the subject. In the late 1920 Richardson [61,62], and Heyting [46] studied possible extensions of the determinant notion to division rings. Heyting defined the designant of a matrix with noncommutative entries, which for 2 × 2 matrices was the Schur complement, and generalized to larger dimensions by induction. Let us stress that both Richardson's and Heyting's quasideterminants were generically rational functions of the matrix coefficients. In 1931, Ore [59] gave a polynomial proposal, the Ore's determinant. A definitive impulse to the modern theory was given by the Gel'fand's school [22,23,[28][29][30][31]. Quasideterminants defined over free division rings were early noticed that are not an analog of the commutative determinant but rather of ratio determinants. An essential aspect for quasideterminants is the heredity principle, quasideterminants of quasideterminants are quasideterminants; there is no analog of such a principle for determinants. Many of the properties of determinants extend to this case, see the cited papers and also [49] for quasi-minors expansions. Already in the early 1990 the Gelf'and school [29] noticed the role quasideterminants for some integrable systems, see also [60] for some recent work in this direction regarding non-Abelian Toda and Painlevé II equations. Nimmo and his collaborators, the Glasgow school, have studied the relation of quasideterminants and integrable systems, in particular we can mention the papers [36][37][38]50]. All this paved the route, using the connection with orthogonal polynomialsà la Cholesky, to the appearance of quasideterminants in the multivariate orthogonality context. Later, in 2006 Olver applied quasideterminants to multivariate interpolation [58].

A.2. Quasideterminants and the heredity principle
Given any partitioned matrix where A i,j ∈ R mi×mj for i, j ∈ {1, . . . , k − 1}, and A k,k ∈ R κ1×κ2 , A i,k ∈ R mi×κ2 and A k,j ∈ R κ1×mj , we are going to define its quasideterminantà la Olver recursively. We start with k = 2, so that A = A1,1 A1,2 A2,1 A2,2 , in this case the first quasideterminant is different to that of the Gel'fand school where Θ 1 (A) = |A| 2,2 = A1,1 A1,2 A2,1 A 2,2 . There is another quasideterminant Θ 2 (A) = A/A 22 = |A| 1,1 = A 1,1 A1,2 A2,1 A2,2 , the other Schur complement, and we need A 2,2 to be an invertible square matrix. Other quasideterminants that can be considered for regular square blocks are A1,1 A1,2 A 2,1 A2,2 and A1,1 A 1,2 A2,1 A2,2 . Following [58] we remark that quasideterminantal reduction is a commutative operation. This is the heredity principle formulated by Gel'fand and Retakh [27,31]: quasideterminants of quasideterminants are quasideterminants. Let us illustrate this by reproducing a nice example discussed in [58]. We consider the matrix and take the quasideterminant with respect to the first diagonal block, which we define as the Schur complement indicated by the non-dashed lines, to get a matrix with blocks with subindexes involving 2 and 3 but not 1. Notice also that we are allowed to take blocks of different sizes we have taken the quasideterminant with respect to a bigger block, composed of two rows and columns of basic blocks. This is the Olver's generalization of Gel'fand's et al. construction. Now, we take the quasideterminant given by the Schur complement as indicated by the dashed lines, to get Θ 2 (Θ 1 (A)) = which is identical to (54), so that Θ I (A) = Θ i1 (Θ i2 (· · · Θ im (A) · · ·)) b In [58], it is defined as the Schur complement with respect to a big block built up by the blocks determined by the indices I.