Rectangular Multiparameter Eigenvalue Problem

• RMEP is a spectral problem defined by rank-drop conditions where rectangular coefficient matrices yield eigenvalue tuples instead of relying on determinants.
• It connects classical multiparameter theory with modern numerical approaches like operator determinants, Macaulay matrix formulations, and minimal-perturbation methods.
• Applications span system identification, differential equation discretizations, and constrained optimization, offering both exact solutions and robust approximations.

Searching arXiv for papers on rectangular multiparameter eigenvalue problems and closely related formulations. The rectangular multiparameter eigenvalue problem (RMEP) is a multiparameter spectral problem in which the coefficient matrices are rectangular, usually with more rows than columns, and an eigenvalue is a parameter tuple at which a multivariate matrix pencil or polynomial loses column rank and admits a nonzero right null vector. In current usage, the term covers both coupled systems

$$(A_{i0}+\lambda_1A_{i1}+\cdots+\lambda_mA_{im})x_i=0$$

with rectangular $A_{ik}$, and the single-equation form

$$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$

or $P(\lambda)y=0$, where rank deficiency rather than determinant vanishing defines the spectrum (Vermeersch et al., 20 May 2026, Vermeersch et al., 20 May 2026).

1. Formal definition and geometric setting

A linear rectangular multiparameter eigenvalue problem is commonly written as

$$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$

with $A_i\in\mathbb C^{k\times \ell}$. In the perturbation-theoretic formulation, the size condition $k=\ell+m-1$ is singled out as the necessary condition for a zero-dimensional solution set; more generally, Macaulay-type formulations assume $k\ge \ell+m-1$ together with generic full normal rank (Vermeersch et al., 20 May 2026, Vermeersch et al., 20 May 2026). An $m$-tuple $\lambda$ is an eigenvalue if and only if

$A_{ik}$0

and an associated right eigenvector $A_{ik}$1 satisfies $A_{ik}$2 (Vermeersch et al., 20 May 2026).

The literature also uses a coupled formulation. In that setting one seeks $A_{ik}$3 and nonzero vectors $A_{ik}$4 such that

$A_{ik}$5

where $A_{ik}$6 are allowed to be rectangular (Vermeersch et al., 20 May 2026). This form is close to classical multiparameter eigenvalue theory, but the decisive distinction is that rectangularity replaces characteristic determinants by rank-drop conditions and makes nullspace geometry central.

Rectangularity changes the local algebra substantially. For a given eigenvalue $A_{ik}$7, right and left nullspaces have different ambient dimensions, and the left nullspace is nontrivial even away from eigenvalues. In the linear theory for simple isolated eigenvalues, the left null space of $A_{ik}$8 has dimension $A_{ik}$9, of which $$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$0 corresponds to the trivial left null space present for all $$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$1 (Vermeersch et al., 20 May 2026). This is one of the defining differences from square one-parameter and square multiparameter problems.

2. Relation to classical multiparameter theory and to rectangular pencils

The classical Atkinson framework concerns coupled square multiparameter systems and their operator determinants. RMEPs inherit much of that algebra only after reformulation. One route, developed for the rectangular case, is to select $$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$2 rows and form square subpencils

$$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$3

whose determinants

$$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$4

define secular equations. At a simple eigenvalue, suitable row selections yield a locally equivalent coupled square MEP, so the rectangular problem can be analyzed through square operator-determinant technology without erasing its rectangular origin (Vermeersch et al., 20 May 2026).

A second route arises from singular two-parameter theory. Košir and Plestenjak describe how linearization of a rectangular two-parameter problem into Atkinson form typically produces singular operator determinants $$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$5, leading to singular generalized eigenvalue problems rather than the nonsingular commuting setting of standard MEPs. In the coprime case for the characteristic polynomials, common regular eigenvalues of the singular GEP pair correspond to genuine eigenpairs of the original singular two-parameter problem; when the characteristic polynomials share a nontrivial common factor, the converse direction remains open and candidate points on the common curve require extra rank and regular-eigenvector tests (Košir et al., 2021).

The rectangular multiparameter matrix pencil problem studied by Jarlebring and coauthors is closely related but not identical. There the starting point is a single rectangular pencil

$$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$6

that loses column rank. For the two-parameter case, the problem is inflated to three simultaneous one-parameter matrix pencil problems by Kronecker commutators

$$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$7

then deflated by symmetry to compressed operators of size

$$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$8

When $$A(\lambda)x=\Big(\sum_{i=0}^m \lambda_i A_i\Big)x=0$$9, the compressed operators are square; the paper proves that the two-parameter rectangular matrix pencil problem always has at least one solution, generically has $P(\lambda)y=0$0 solutions, and under a rank assumption becomes equivalent to three simultaneous commuting eigenvalue problems (Gungah et al., 2024). This establishes a precise bridge between single-pencil rectangular rank-loss problems and the commuting-eigenproblem ideology of RMEP.

3. Existence, finiteness, and spectral structure

Generic counting results are available for important classes. For a generic linear RMEP with matrices of size $P(\lambda)y=0$1, the number of eigenvalues is

$P(\lambda)y=0$2

For a generic polynomial RMEP of degree $P(\lambda)y=0$3 and normal rank $P(\lambda)y=0$4, the number of eigenvalues is

$P(\lambda)y=0$5

and all are finite in the generic case (Hochstenbach et al., 2022). These counts are derived by reducing the problem to standard MEP machinery or by a Bézout-type intersection argument involving the degree of the homogenized image variety and the determinantal rank-deficiency locus.

These generic counts do not imply that every rectangular problem has exact solutions. The minimal-perturbation paper makes the opposite point explicit: because each equation is overdetermined, an RMEP may have no exact solution in its original form. This is not a pathology of a particular algorithm but a structural consequence of rectangularity. The proposed remedy is to define approximate eigen-tuples through the smallest Frobenius-norm perturbation that renders the perturbed problem exactly solvable (Han et al., 8 Aug 2025).

The geometry of the solution set can also be non-isolated. MacaulayLab is designed to handle the case in which the affine solution set is zero-dimensional while positive-dimensional components occur only at infinity; the toolbox identifies affine solutions through block-wise rank structure and column compression of Macaulay nullspaces (Vermeersch et al., 20 May 2026). By contrast, in singular two-parameter linearizations, common-factor curves may create non-isolated candidate sets whose interpretation as genuine eigenvalues is more delicate (Košir et al., 2021).

A common misconception is that rectangularity merely enlarges the equation count while leaving the underlying spectral picture unchanged. The recent literature shows the opposite: exact solvability may fail, operator determinants may be singular, left nullspaces are generically nontrivial, and even when an equivalent square MEP exists, it is often only local or obtained after nontrivial lifting, compression, or regularization (Vermeersch et al., 20 May 2026, Han et al., 8 Aug 2025).

4. Numerical solution strategies

One major strategy is reduction to standard MEPs. For linear RMEPs, Hochstenbach, Košir, and Plestenjak propose random or deterministic projection matrices $P(\lambda)y=0$6 that turn one rectangular problem into a square $P(\lambda)y=0$7-parameter MEP. The resulting operator determinants can be compressed from $P(\lambda)y=0$8 to the optimal symmetric-tensor dimension

$P(\lambda)y=0$9

eliminating redundant solutions. For structured quadratic two-parameter problems arising in ARMA and LTI identification, they develop tailored linearizations and Vandermonde-type compressions, reducing dimensions from $$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$0 to $$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$1 in the generic quadratic case and yielding significantly smaller singular commuting GEPs than block Macaulay constructions (Hochstenbach et al., 2022).

A second strategy is the block Macaulay approach. MacaulayLab treats an RMEP as a polynomial matrix equation

$$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$2

and builds structured block Macaulay matrices $$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$3. Null-space or column-space extraction then yields multiplication maps of the form

$$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$4

whose joint eigenstructure returns the parameter tuples. The package is described as basis- and monomial-order independent, and it is explicitly designed to cope with positive-dimensional solution sets at infinity through block-wise rank checks and column compression (Vermeersch et al., 20 May 2026).

In optimization-driven identification problems, the RMEP can appear directly as a KKT characterization. In the fixed-pole least-squares realization problem, De Moor, Vermeersch, and collaborators derive a cubic rectangular matrix polynomial

$$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$5

whose affine eigenvalues $$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$6 enumerate all critical points of the constrained problem; local and global minimizers form a subset of those affine eigenvalues. The paper points to MacaulayLab and MultiParEig as practical solvers for this RMEP reformulation (Vermeersch et al., 11 Sep 2025).

Once a solver has produced a commuting family, randomized postprocessing can improve joint-eigenvalue extraction. The randomized methods paper proposes choosing a random linear combination of commuting or nearly commuting matrices, computing its eigenvectors, and then recovering the joint eigenvalues through one-sided or two-sided Rayleigh quotients. In MultiParEig 2.8 this replacement of Schur-based clustering improved accuracy and reliability for multiparameter problems, and the same mechanism applies to RMEP reductions that produce commuting multiplication matrices or commuting operator-determinant quotients (He et al., 2024).

5. Perturbation theory, conditioning, and approximate eigen-tuples

The first systematic perturbation theory for the linear rectangular problem introduces norm-wise backward errors, condition numbers, and pseudospectra adapted to the fact that $$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$7 is rectangular and that left and right eigenvectors have different dimensions. For an approximate eigenpair $$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$8, the eigenpair backward error is

$$A(\lambda)x = A_0x+\sum_{i=1}^m \lambda_i A_i x = \sum_{i=0}^m \lambda_i A_i x = 0, \qquad \lambda_0=1,$$9

and the eigenvalue-only backward error is

$A_i\in\mathbb C^{k\times \ell}$0

For a simple eigenvalue $A_i\in\mathbb C^{k\times \ell}$1, the eigenvalue condition number is

$A_i\in\mathbb C^{k\times \ell}$2

where $A_i\in\mathbb C^{k\times \ell}$3 is assembled from a basis of the left null space through entries $A_i\in\mathbb C^{k\times \ell}$4. The $A_i\in\mathbb C^{k\times \ell}$5-pseudospectrum is characterized by the singular-value inequality

$A_i\in\mathbb C^{k\times \ell}$6

so backward error and pseudospectrum coincide exactly in the linear rectangular setting (Vermeersch et al., 20 May 2026).

For two-parameter problems with $A_i\in\mathbb C^{k\times \ell}$7 and $A_i\in\mathbb C^{k\times \ell}$8, the same paper relates the Jacobian of the secular equations to the auxiliary matrix $A_i\in\mathbb C^{k\times \ell}$9. The consequence is geometric: nearly parallel intersections of secular curves correspond to near-singularity of $k=\ell+m-1$0 and hence to large $k=\ell+m-1$1. This gives a direct link between local algebraic geometry and spectral sensitivity (Vermeersch et al., 20 May 2026).

When no exact eigen-tuple exists, the minimal-perturbation framework supplies an optimization-based surrogate. In homogeneous variables $k=\ell+m-1$2, with $k=\ell+m-1$3 and $k=\ell+m-1$4, the objective becomes

$k=\ell+m-1$5

subject to $k=\ell+m-1$6. The alternating algorithm updates each $k=\ell+m-1$7 as the smallest right singular vector of

$k=\ell+m-1$8

and then updates $k=\ell+m-1$9 as the eigenvector associated with the smallest eigenvalue of

$k\ge \ell+m-1$0

The paper states that the scheme monotonically decreases the objective and converges to a stationary point satisfying the KKT conditions (Han et al., 8 Aug 2025).

An empirical observation reported in the perturbation paper is that, in system identification problems with RMEP reformulations, globally optimal solutions tend to coincide with the best-conditioned eigenvalues. The authors present this as a conjecture rather than a theorem, which is significant because it distinguishes a numerical regularity phenomenon from an established structural law (Vermeersch et al., 20 May 2026).

6. Applications, interpretations, and current directions

System identification is the most developed application domain in the recent RMEP literature. The 2022 numerical-methods paper shows that stationary points of optimal least-squares ARMA and autonomous LTI fitting can be formulated as polynomial RMEPs, with all stationary points—including the global minimum—recoverable from compressed singular commuting GEPs (Hochstenbach et al., 2022). The fixed-pole realization paper refines this by incorporating prescribed poles into the model factorization $k\ge \ell+m-1$1, thereby reducing the unknown polynomial degree and the size of the multiparameter problem; in the reported experiments, the fixed-pole formulation led to $k\ge \ell+m-1$2 affine eigenvalues in one constrained case and $k\ge \ell+m-1$3 solutions in another, compared with $k\ge \ell+m-1$4 solutions for the unconstrained setting (Vermeersch et al., 11 Sep 2025).

Differential-equation discretizations provide a second major source. The minimal-perturbation paper validates its methods on least-squares spectral discretizations of multiparameter Sturm–Liouville equations and of a Helmholtz problem in elliptical coordinates, both of which produce rectangular algebraic systems of the form

$k\ge \ell+m-1$5

In the closed-form Sturm–Liouville example with $k\ge \ell+m-1$6, $k\ge \ell+m-1$7, $k\ge \ell+m-1$8, the exact eigenvalues are

$k\ge \ell+m-1$9

and the truncated-SVD reduction returns highly accurate approximations (Han et al., 8 Aug 2025).

There are also neighboring application areas in which singular and rectangular multiparameter tools appear even when the final formulation is not strictly rectangular. The layered leaky-wave paper casts half-space-coupled wave propagation into singular four- and six-parameter MEPs with square per-equation blocks, then solves them using operator determinants, staircase regularization, and shifted-$m$0 techniques explicitly associated in the paper with singular/rectangular multiparameter theory (Gravenkamp et al., 2024). Conversely, the nonlinear eigenvalue paper on quadratic rational eigenvector nonlinearities produces spurious spectra that are themselves rectangular two-parameter eigenproblems,

$m$1

and uses structure-preserving Arnoldi projections to filter those RMEP-induced modes (Janssens et al., 3 Oct 2025).

The current research front is therefore not a single solver or a single definition, but a cluster of interlocking viewpoints: direct rectangular rank-drop formulations, square multiparameter liftings, singular operator-determinant analysis, Macaulay-based polynomial elimination, perturbation theory, and minimal-perturbation approximation. Taken together, these developments indicate that the modern RMEP is best understood not as a minor variant of square MEPs, but as a distinct spectral framework whose defining feature is the interaction between multiparameter algebra and rectangular rank geometry (Vermeersch et al., 20 May 2026, Vermeersch et al., 20 May 2026).