On a Solution of the Multidimensional Truncated Matrix-Valued Moment Problem

We will consider the multidimensional truncated p×p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times p$$\end{document} Hermitian matrix-valued moment problem. We will prove a characterisation of truncated p×p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times p$$\end{document} Hermitian matrix-valued multisequence with a minimal positive semidefinite matrix-valued representing measure via the existence of a flat extension, i.e., a rank preserving extension of a multivariate Hankel matrix (built from the given truncated matrix-valued multisequence). Moreover, the support of the representing measure can be computed via the intersecting zeros of the determinants of matrix-valued polynomials which describe the flat extension. We will also use a matricial generalisation of Tchakaloff’s theorem due to the first author together with the above result to prove a characterisation of truncated matrix-valued multisequences which have a representing measure. When p=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = 1$$\end{document}, our result recovers the celebrated flat extension theorem of Curto and Fialkow. The bivariate quadratic matrix-valued problem and the bivariate cubic matrix-valued problem are explored in detail.


Introduction
In this paper, we will investigate the multidimensional truncated matrix-valued moment problem. Given a truncated multisequence S = (S γ ) 0≤|γ|≤m , where S γ ∈ H p (i.e., S γ is a p × p Hermitian matrix), we wish to find necessary and sufficient conditions on S for the existence of a p × p positive matrix-valued measure T on R d , with convergent moments, such that for all γ = (γ 1 , . . . , γ d ) ∈ N d 0 such that 0 ≤ |γ| ≤ m. We would also like to find a positive matrix-valued measure T = κ a=1 Q a δ w (a) on R d such that (1.1) holds and and κ a=1 rank Q a is as small as possible, (1.2) i.e., T is a finitely atomic measure of the form T = κ a=1 δ w (a) Q a with κ a=1 rank Q a = rank M (n) (confer Remark 3.4). If (1.1) holds, then T is called a representing measure for S. If (1.1) and (1.2) are in force, then T is called a minimal representing measure for S.
Before proceeding any further, we will first introduce frequently used notation. Commonly used sets are N 0 , R, C denoting the sets of nonnegative integers, real numbers and complex numbers respectively. Given a nonempty set E, we let Next, we let C p×p denote the set of p × p matrices with entries in C and H p ⊆ C p×p denote the set of p×p Hermitian matrices with entries in C. Given x = (x 1 , . . . , x d ) ∈ R d and λ = (λ 1 , . . . , λ d ) ∈ N d 0 , we define x λ j j and |λ| = λ 1 + · · · + λ d and Γ m,d := {γ ∈ N d 0 : 0 ≤ |γ| ≤ m}. Throughout the entirety of this paper we will assume that the given H p -valued truncated multisequence S = (S γ ) γ∈Γ 2n,d satisfies (1.3) Let us justify this assumption. If S 0 d 0 (i.e., S 0 d is positive definite), then we can simply replace S by S = ( S γ ) γ∈Γ 2n,d , where S γ = S , S 0 d is positive semidefinite) and not invertible, then Lemma 5.50 and Smuljan's lemma (see Lemma 2.1) readily show that we must necessarily have that Ran S γ ⊆ Ran S 0 d and hence KerS 0 d ⊆ KerS γ for all γ ∈ Γ 2n,d . Consequently, we can find a unitary matrix U ∈ C p×p such that U * S γ U = S γ 0 0 0 and we may replace S by S = ( S γ ) γ∈Γ 2n,d , where S γ ∈ Cp ×p withp = rank S 0 d , and normalise as above.

Main Contributions
(C1) We will characterise positive infinite d-Hankel matrices based on a H p -valued multisequence via an integral representation. Indeed, we will see that S (∞) = (S γ ) γ∈N d 0 gives rise to a positive infinite d-Hankel matrix M (∞) with finite rank if and only if there exists a finitely atomic positive H p -valued measure T on R d such that In this case, the support of the positive H p -valued measure T agrees with V(I), where V(I) is the variety of a right ideal of matrix-valued polynomials based on the kernel of M (∞) (see Definition 5.20) and the cardinality of the support of T is exactly rank M (∞) (see Theorem 5.65). (C2) Let S = (S γ ) γ∈Γ 2n,d be a given truncated H p -valued multisequence. We will see that S has a minimal representing measure T = Mädler [33] studied the truncated matrix-valued Stieltjes moment problem via a similar approach.
Bakonyi and Woerdeman in [5] studied the univariate truncated matrix-valued Hamburger moment problem and the odd case of the bivariate truncated matrixvalued moment problem. The first author and Woerdeman in [53] investigated the odd case of the truncated matrix-valued K-moment problem on R d , C d and T d , where they discovered easily checked commutativity conditions for the existence of a minimal representing measure.
Applications of matrix-valued moment problems and related topics have been studied extensively in recent years. Geronimo [39] studied scattering theory and matrix orthogonal polynomials with the construction of a matrix-valued distribution function built from matrix-valued moments. Dette and Studden in [27] investigated matrix orthogonal polynomials and matrix-valued measures associated with certain matricial moments from a numerical analysis point of view. In [28], Dette and Studden considered optimal design problems in linear models as a statistical application of the problem of maximising matrix-valued Hankel determinants built from matricial moments. Moreover, Dette and Tomecki in [29] studied the distribution of random Hankel block matrices and random Hankel determinant processes with respect to certain matricial moments.

Structure
The paper is organised as follows. In Sects. 2, 3 and 4, we will formulate a number of definitions and basic results for future use. In Sect. 5.1, we will define infinite d-Hankel matrices and prove a number of results on the right ideal of matrix Vol. 90 (2022) On the truncated matrix-valued moment problem 23 polynomials beloning to the kernel of a d-Hankel matrix. In Sect. 5.2 we will show that every H p -valued multisequence which gives rise to a positive infinite d-Hankel matrix with finite rank has a representing measure. In Sect. 5.3, we will prove a number of necessary conditions for a H p -valued multisequence to have a representing measure. In Sect. 5.4, we will precisely formulate and proof (C1). In Sect. 5.5, we will formulate a lemma which states that once a d-Hankel matrix has a flat extension, one can construct a sequence of flat extensions of all orders giving rise to positive infinite d-Hankel matrix with finite rank. In Sect. 6, we will prove our flat extension theorem, i.e., (C2). In Sect. 7, we will prove an abstract characterisation of truncated H p -valued multisequences with a representing measure, i.e., (C3). In Sect. 8.1, we will provide necessary and sufficient conditions for the bivariate quadratic matrix-valued moment problem to have a minimal solution. In Sect. 8.2, we will analyse the bivariate quadratic matrix-valued moment problem when the 2-Hankel matrix M (1) is block diagonal. In Sect. 8.3, we will consider some singular cases of the bivariate quadratic matrix-valued moment problem. In Sect. 8.4, we will see that the bivariate quadratic matrix-valued moment problem has a minimal solution whenever the corresponding 2-Hankel matrix M (1) is positive definite and satisfies a certain condition which automatically holds if p = 1, i.e., the first part of (C4). In Sect. 8.5 we will go through a number of examples for the bivariate quadratic matrix-valued moment problem. In particular, we will see that S 00 = I p and M (1) 0 is not enough to guarantee that a minimal solution exists. Finally, in Sect. 9, we will consider a particular case of the bivariate cubic matrix-valued moment problem. For the convenience of the reader we have compiled a list of commonly used notation that appears throughout the paper.

Notation
R m×n and C m×n denote the vector spaces of real and complex matrices of size m × n, respectively. We will let H p denote the real vector space of p × p Hermitian matrices in C p×p .
The p × p identity matrix will be denoted by I p or I (when no confusion can possibly arise) and the p × p matrix of zeros will be denoted by 0 p×p or 0 (when no confusion can possible arise).
C p denotes the p-dimensional complex vector space equipped with the standard inner product ξ, η = η * ξ, where ξ, η ∈ C p . The standard basis vectors in C p will be denoted by e 1 , . . . e p .
A * and A T denote the conjugate transpose and transpose, respectively, of a matrix A ∈ C n×n . If A ∈ C n×n is invertible, then we will let A − * := (A −1 ) * = (A * ) −1 .
Let M ∈ C m×n . Then C M and ker(M ) denote the column space and null space, respectively.
σ(A) denotes the spectrum of a matrix A ∈ C n×n . We will write A 0 (resp. A 0) if A is positive semidefinite (resp. positive definite).
col(C λ ) λ∈Λ and row(C λ ) λ∈Λ denote the column and row vectors with entries (C λ ) λ∈Λ , respectively. N d 0 , R d and C d denote the set of d-tuples of nonnegative integers, real numbers and complex numbers, respectively.
V(M (n)) will denote the variety of the d-Hankel matrix M (n) (see Definition 3.5).
B(R d ) will denote the sigma algebra of Borel sets on R d . card Ω denotes the cardinality of the set Ω ⊆ R d .

Preliminaries
In this section we shall provide preliminary definitions and results for future use.
We will begin with a useful characterisation for positive extensions given by Smuljan [76] via the following result.
Then the following statements hold: (i)Ã is positive semidefinite if and only if B = AW for some W ∈ C n×m and C W * AW. (ii)Ã is positive semidefinite and rankÃ = rank A if and only if B = AW for some W ∈ C n×m and C = W * AW.
Vol. 90 (2022) On the truncated matrix-valued moment problem 25 The support of an H p -valued measure T, denoted by supp T, is defined as the for all u, v ∈ C p , provided all integrals on the right-hand side converge. on R d are given by for all u, v ∈ C p , provided all integrals on the right-hand side converge, that is, where T ab is as in Definition 2.3.
then it is easy to check that Definition 2.9. The power moments of a positive H p -valued measure T on R d are given by Definition 2.10. Given distinct points w (1) , . . . , w (k) ∈ R d and a subset Λ = {λ (1) , . . . , Vol. 90 (2022) On the truncated matrix-valued moment problem 27 We now present [53,Theorem 2.13] which is based on [69, Algorithm 1] and provides a useful machinery when the invertibility of a multivariable Vandermonde matrix is needed to compute the weights of a representing measure. Theorem 2.11. Given distinct points w (1) , . . . , w (κ) ∈ R d , there exists Λ ⊆ N d 0 such that card Λ = κ and V (w (1) , . . . , w (κ) ; Λ) is invertible.

d-Hankel Matrices
In this section we will define d-Hankel matrices and the variety of a d-Hankel matrix. We next define the variety of a d-Hankel matrix in our matrix-valued setting. We introduce zeros of determinants of matrix-valued polynomials abstracting that way the notion of the variety of a d-Hankel matrix which can implicitly apeared first in Curto and Fialkow [16].
The variety of M (n), denoted by V(M (n)), is given by

Matrix-Valued Polynomials
We introduce important definitions and notation while establishing several algebraic results involving matrix-valued polynomials with several real indeterminates which will be important for proving our flat extension theorem for matricial moments.
Remark 4.4. We wish to justify the usage of the moniker real radical of Definition 4.3 when p = 1. We note that one usually says that a real ideal (see, e.g., [60]). Suppose I = I 1 + I 2 i, where On the truncated matrix-valued moment problem 29 and let f (a) = q (a) + r (a) i, where q (a) (x) = Re(f (x)) and r (a) (x) = Im(f (x)).
In the following remark we will introduce an additional assumption on I ⊆ C[x 1 , . . . , x d ] which appears in Remark 4.4. As we noted in Remark 4.4, Thus, it is clear that f ∈ I vanishes on a set V ⊆ R d if and only if Re(f (x)) and ± Im(f (x)) vanish on V. In view of the Real Nullstellensatz (see, e.g., [9]), any real radical ideal must agree with its vanishing ideal (that is, the set of polynomials which vanish on the variety). Therefore, if I ⊆ C[x 1 , . . . , x d ] is real radical, then f ∈ I implies thatf ∈ I . Since I is an ideal in C[x 1 , . . . , x d ] which is closed under complex conjugation, we have that I 1 and I 2 are subideals of I over R[x 1 , . . . , x d ]. Hence, we may use the fact that I is real radical to deduce (i) and (ii).
Proof. We proceed by induction on p.
Suppose the claim holds for p > 2. We have Let L(x) be the sum of the terms of det P (x) of degree up to γ(p − 1) with |γ| > 0 and C(x) the sum of the terms of det P (x) of degree up to γp with |γ| = 0. Then where m < |γ|p. Thus We order the monomials in C[x 1 , . . . , x d ] by the graded lexicographic order ≺ grlex .
where γp := (γ 1 p, . . . , γ d p) ∈ N d 0 and m < |γ|p, the leading term of ϕ(x) is LT(ϕ(x)) = x γp . Definition 4.8. We define the basis of C p×p viewed as a vector space over C where E jk ∈ C p×p is the matrix with 1 in the (j, k)-th entry and 0 in the rest of the entries, j, k = 1, . . . , p.
where E jk ∈ C p×p is as in Definition 4.8 for all j, k = 1, . . . , p. Proof. We need to show Without loss of generality, we may assume that and I jj is real radical.

Positive Infinite d-Hankel Matrices with Finite Rank
We shall study positive infinite d-Hankel matrices with finite rank, necessary conditions for a truncated H p -valued multisequence to have a representing measure and extension results for positive d-Hankel matrices.

Infinite d-Hankel Matrices
In this subsection we define d-Hankel matrices associated with an H p -valued multisequence. We investigate positive infinite d-Hankel matrices with finite rank and a right ideal of matrix-valued polynomials generated by column relations.
0 is a right module over C p×p , under the operation of addition given by together with the right multiplication given by Proof. The verification that (C p×p ) ω 0 is a right module over C p×p can be carried out in a very straight-forward manner.
We now give the definition of an infinite d-Hankel matrix based on S (∞) : We define M (∞) to be the corresponding moment matrix based on S (∞) as follows. We label the block rows and block columns by a family of monomials (x γ ) γ∈N d 0 ordered by ≺ grlex . We let the entry in the block row indexed by x γ and in the block column indexed by xγ be given by For S := (S γ ) γ∈Γ 2n,d a given truncated H p -valued multisequence and M (n) the corresponding d-Hankel matrix, we let X λ := col(S λ+γ ) γ∈Γ n,d for λ ∈ Γ n,d and C M (n) be the column space of M (n).

Indeed, notice that
We will write M (∞) 0 if Vol. 90 (2022) On the truncated matrix-valued moment problem 35 or, equivalently, M (n) 0 for all n ∈ N d 0 .
Definition 5.11. Let C p [x 1 , . . . , x d ] be the set of vector-valued polynomials, that is, Then by Definition 5.10, If e 1 is a standard basis vector in C p , then We define the set and the kernel of the map Φ : Thus, all singular values of A are 0 and so rank A = 0, which forces We write Since M ( ) 0, by Lemma Thus, Eq. (5.2) holds for all ≥ m and we obtain We have to show the following: To prove (ii) we need to show that if P ∈ ker Φ and Q ∈ C p×p [x 1 , . . . , x d ], then We will show We have Finally, since as desired and we derive that ker Φ is a right ideal. By Lemma 5.14, I = ker Φ and so I is a right ideal as well.
Definition 5.16. Suppose M (∞) 0 and let I be as in Definition 5.13. We define the right quotient module of equivalence classes modulo I, that is, we will write Vol. 90 (2022) On the truncated matrix-valued moment problem 39 . , x d ]/I is a right module over C p×p , under the operation of addition (+) given by , together with the right multiplication (·) given by The following properties can be easily checked: (iv) (P + I)I p = P + I.
given by The following lemma shows that the form in Definition 5.18 is a well-defined positive semidefinite sesquilinear form. Proof. We first show that the form [P + I, Q + I] is well-defined. We need to prove that if P + I = P + I and Q + I = Q + I, then Since P − P ∈ I, We write We sum both hand sides of Eqs. (5.4) and (5.5) and we obtain We now show that [P + I, In analogy to Definition 3.5, we define the variety associated with the right ideal I.
We will show Notice that We write Then by Eq. (5.6). Thus, Eq. (5.7) holds and the proof is complete.
The following lemma is well-known, see, e.g., Horn and Johnson [47]. However, for the convenience of the reader, we provide a statement. Lemma 5.22. Let A ∈ C n×n and B ∈ C m×m be given. Then

Existence of a Representing Measure for a Positive Infinite d-Hankel Matrix
with Finite Rank In this subsection we shall see that if M (∞) 0 and rank M (∞) < ∞, then the associated H p -valued multisequence has a representing measure T.
Definition 5.24. We define the vector space Proof. If dimC M (∞) = m and m = r, then there exists a basis where e k a is a standard basis vector in C p and a = 1, . . . , m. We will show thatB Let Then by Eq.
However, this contradicts the fact that B is linear independent. HenceB is linearly , where e k a is a standard basis vector in C p and a = 1, . . . , r.
Then there exist c 1 , . . . , c r ∈ C such that any w ∈C M (∞) can be written as In analogy to results from Section 3.1, we move on to the following. where and the kernel of the map φ Vol. 90 (2022) On the truncated matrix-valued moment problem 45 Moreover, since M (∞) 0, M(m) 0 and hence, the square root of M (m) exists.
0 and J be as in Definition 5.32. We define the quotient space Proof. We first show that the inner product h + J , q + J is well-defined. We need to prove that if h + J = h + J and q + J = q + J , then We sum both hand sides of Eqs. (5.10) and (5.11) and we obtain We now show that the inner product h + J , q + J is linear. We must prove that Then We may view q as q(x) = λ∈Γ n,d q λ x λ and we have Finally, we show h + J , q + J is positive semidefinite. By definition, Since M (∞) 0, by Lemma 5.12, Hence h + J , h + J is positive semidefinite.
Vol. 90 (2022) On the truncated matrix-valued moment problem 47 Definition 5.37. We define the map Ψ : Lemma 5.38. Ψ as in Definition 5.37 is an isomorphism.
Proof. We consider the map φ : Moreover, we shall see that φ is surjective. Indeed, for every By the Fundamental homomorphism theorem (see, e.g., [ In this setting, we present the multiplication operators M x j , j = 1, . . . , d, as defined below.
We define the multiplication operators or equivalently, which is equivalent to that is, and hence x j (q − h) ∈ J as required. Proof. For all j = 1, . . . , d, that is, wheref ∈ (C p ) ω 0 and (x j q) ∈ (C p ) ω 0 and equation (5.13) is equal to and the proof is now complete.
Next, we shall use spectral theory involving the preceding multiplication operators. First, we denote by P the set of the orthogonal projections on C p [x 1 , . . . , x d ]/J .
E j is unique, in the sense that if F j : B(R) → P is another spectral measure such that Vol. 90 (2022) On the truncated matrix-valued moment problem 51 By [71,Lemma 4.3], E j (α)E j (β) = E j (α ∩ β) for α, β ∈ B(σ(E j )), which implies that Since M x j is self-adjoint and pairwise commute, that is, Thus, by [71,Theorem 4.10], there exists a unique spectral measure E on the Borel algebra B(Ω) of the product space Ω We next fix a basis D of C p [x 1 , . . . , x d ]/J and let A j ∈ C r×r be the matrix representation of M x j with respect to D. Then since M x j are commuting self-adjoint operators we get  The following proposition proves the existence of a representing measure T for a given H p -valued multisequence S (∞) := (S γ ) γ∈N d 0 which gives rise to an infinite d-Hankel matrix with finite rank. In Sect. 5.4 we will obtain additional information on the representing measure T .
Therefore, we have obtained the left hand side of the equation (5.14). The right hand side is implied by Remark 5.46. Indeed we have Fix α ∈ B(R d ) and define Vol. 90 (2022) On the truncated matrix-valued moment problem 53 We observe by the Rayleigh-Ritz Theorem (see, e.g., [47,Theorem 4 by the Cauchy-Schwarz inequality. Hence β is a bounded sesquilinear form. For By the Riesz Representation Theorem for Hilbert spaces (see, e.g., [62, Theorem 4, Section 6.3]), there exists a unique ϕ ∈ C p such that for which T (α)w = ϕ, α ∈ B(R d ).

Necessary Conditions for the Existence of a Representing Measure
Throughout this subsection a series of lemmas are shown on the variety of the d-Hankel matrix and its connection with the support of the representing measure. Proof. For η = col(η λ ) λ∈Γ n,d , we have Definition 5.51. Let T be a representing measure for S : P (w (a) ) * Q a P (w (a) ).
The following lemma will connect the support of a representing measure of an H p -valued truncated multisequence and the variety of the d-Hankel matrix M (n) and is a matricial generalisation of Proposition 3.1 in [16] (albeit in an equivalent complex moment problem setting). As we will see in Example 5.54, unlike the scalar setting (i.e., when p = 1), we only have one direction of the implication. Moreover, the proof of Lemma 5.53 is more cumbersome than the scalar case.
Suppose to the contrary that supp T Z(det P (x)).
Then there exists a point u (0) ∈ supp T such that u (0) / ∈ Z(det P (x)) and has the property T (B ε (u (0) )) = 0 p×p and B ε (u (0) ) ∩ Z(det P (x)) = ∅. We write and we note that both terms on the right hand side are positive semidefinite.
We continue with results on the variety of a d-Hankel matrix and its connection with the support of a representing measure T.
Consider P (X) ∈ C M (n) . We have Z(det P (x)) = supp T and we shall see P (X) = col(0 p×p ) γ∈Γ n,d . We notice and hence P (X) = col(0 p×p ) γ∈Γ n,d . Since there exists matrix-valued polynomial P (x) such that P (X) = col(0 p×p ) γ∈Γ n,d and Z(det P (x)) = supp T, we then have Next, for the choice of P ( We have Z(det P (x)) = supp T and we consider P (X) ∈ C M (n) . We need to show that for this choice of P (x), P (X) = col(0 p×p ) γ∈Γ n,d . Notice that and so P (X) = col(0 p×p ) γ∈Γ n,d . We thus conclude that there exists a matrix-valued polynomial P (x) such that P (X) = col(0 p×p ) γ∈Γ n,d and Z(det P (x)) = supp T and thus we obtain rank Q a and the proof is complete. where By Sylvester's law of inertia (see, e.g., [ Then we shall obtain det which implies that Z(det P (x)) ⊆ {w (1) , . . . , w (κ) }. Thus In the following, we shall see that for both choices of the matrix-valued polynomial P ∈ C p×p [x 1 , . . . , x d ], one obtains P ∈ I and this in turn yields the inclusion V(I) ⊆ supp T. Consider first the matrix-valued polynomial P ( as well. Consider P (X) ∈ C M (∞) . We have Z(det P (x)) = supp T and we shall see and so P ∈ I. Since there exists P ∈ I such that Z(det P (x)) = supp T, we again obtain V(I) ⊆ supp T as desired. In the next lemma we treat the multiplication operators of Definition 5.40 to provide a connection between the joint spectrum of M x 1 , . . . , M x d and a representing measure T.

Characterisation of Positive Infinite d-Hankel Matrices with Finite Rank
In this subsection we will characterise positive infinite d-Hankel matrices with finite rank via an integral representation. Moreover, we will connect the variety of the associated right ideal of the d-Hankel matrix with the support of the representing measure and also make a connection between the rank of the positive infinite d-Hankel matrix and the cardinality of the support of the representing measure. We next state and prove the main theorem of this section. We shall see that if M (∞) 0 with rank M (∞) < ∞, then the associated H p -valued multisequence has a unique representing measure T and one can extract information on the support of the representing measure in terms of the variety of the right ideal associated with M (∞).

Positive Extensions of d-Hankel Matrices
We investigate positive extensions of a d-Hankel matrix based on a truncated H pvalued multisequence. Both results provided in this subsection are important for obtaining the flat extension theorem for matricial moments stated and proved in Sect. 6. Proof. See Appendix A for a proof.

The Flat Extension Theorem for a Truncated Matricial Multisequence
In this section, we will formulate and prove our flat extension theorem for matricial moments. We will see that a given truncated H p -valued multisequence S := (S γ ) γ∈Γ 2n,d has a minimal representing measure (see For the convenience of the reader, please note that Assumption 1.3 is in force.

Abstract Solution for the Truncated H p -Valued Moment Problem
In this section we will formulate an abstract criterion for a truncated H p -valued multisequence to have a representing measure. Proof. If S has an eventual extension M (n + k) such that M (n + k) admits a flat extension, then we may use Theorem 6.2 to see that S has a representing measure. Conversely, if S has a representing measure, then we may use the first author's H p -valued generalisation of Bayer and Teichmann's generalisation of Tchakaloff's theorem (see [51]) to see that we can always find a finitely atomic representing measure for S. One can argue much in the same way as in the proof of Theorem 6.2 to see that M (n) has an eventual extension M (n + k) which in turn has a flat extension.
We shall see that a direct analogue of Curto and Fialkow's results on the bivariate quadratic moment problem does not hold when p ≥ 2 (see Example 8.12). However, we shall see that if M (1) is positive semidefinite and certain block column relations hold, then S = (S γ ) γ∈Γ 2,2 , S 00 0, has a minimal representing measure.

General Solution of the Bivariate Quadratic Matrix-Valued Moment Problem
The next theorem illustrates necessary and sufficient conditions for a given quadratic H p -valued bisequence to have a minimal representing measure. We observe that the positivity and flatness conditions are key to obtain a minimal solution to the bivariate quadratic matrix-valued moment problem.    We derive the matrix equations (8.1), (8.2) and (8.3), respectively.

The Block Diagonal Case of the Bivariate Quadratic Matrix-Valued Moment Problem
In the next theorem we shall see that every truncated H p -valued bisequence S := (S γ ) γ∈Γ 2,2 with M (1) 0 being block diagonal has a minimal representing measure.
In what follows, given A, B ∈ C n×n , we shall let σ(A, B) denote the set of generalised eigenvalues of A and B, i.e., The reader is encouraged to see [40] for results on generalised eigenvalues.
The following example showcases Theorem 8.8 for an explicit truncated H 2valued bisequence.
In what follows, we shall say that T = κ a=1 Q a δ (x a ,y a ) is a minimal representing measure for S = (S γ ) γ∈Γ 3,2 if κ a=1 rankQ a = rankM (1). In our matricial setting, we have the following result. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.