- The paper develops a perturbation framework for rectangular multiparameter eigenvalue problems, including sharp norm-wise backward errors and computable eigenvalue and eigenvector condition numbers for isolated, simple solutions.
- The paper characterizes the multiparameter pseudospectrum through residuals, smallest singular values, and pseudoinverses, and introduces slice-wise algorithms for computing it when simultaneous matrix reduction is unavailable.
- The paper links poor conditioning to nearly tangential secular-curve intersections and reports evidence that globally optimal system-identification solutions tend to be among the best-conditioned eigenvalues, although this claim remains conjectural.
Problem setting and motivation
The paper develops a perturbation theory for the rectangular multiparameter eigenvalue problem (rMEP): given coefficient matrices $\mat_i \in \Cset^{k \times \ell}$ with k=ℓ+m−1 and full normal rank, find tuples $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$ and nonzero vectors $\rvec$ such that
$\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$
Eigenvalues are equivalently characterized as points where the rank of $\mep{\eig}$ drops below its normal rank, or as common roots of the L=(ℓk) secular equations obtained from determinants of all ℓ-row selections. The size condition k=ℓ+m−1 is necessary (but not sufficient) for a zero-dimensional solution set; for one parameter it reduces to square coefficient matrices. The analysis is restricted to isolated, finite, simple solutions of linear problems.
The rectangular shape obstructs direct reuse of existing theory: at an eigenvalue the left null space has dimension m, containing k=ℓ+m−10 trivial polynomial vectors plus one nontrivial left eigenvector, and left and right eigenvectors live in spaces of different dimension. Prior work by Hochstenbach and Plestenjak covered coupled square multiparameter problems in 2003; this paper extends backward error analysis, conditioning, and pseudospectra to the rectangular single-equation setting (2605.21013).
Backward error analysis
For an approximate eigenpair k=ℓ+m−11 with unit-norm eigenvector, perturbations are bounded norm-wise via error matrices k=ℓ+m−12 (absolute: k=ℓ+m−13; relative: k=ℓ+m−14). The norm-wise backward eigenpair error admits the closed form
k=ℓ+m−15
where k=ℓ+m−16 is the residual and k=ℓ+m−17. This generalizes the Frayssé–Higham results for generalized eigenvalue problems (GEP) to k=ℓ+m−18 parameters, and reduces exactly to them when k=ℓ+m−19. Attaining perturbations are rank-one matrices built from a dual vector of $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$0.
When only eigenvalues matter, the optimal backward error minimizes over all eigenvectors:
$\eig = (\eigcomp[1], \ldots, \eigcomp[m])$1
via the standard singular value characterization. In the running example, the pair error is $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$2 while the optimal eigenvalue error is $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$3, confirming that discarding the eigenvector can only lower the measured backward error. A corollary identifies the set $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$4 with the $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$5-pseudospectrum, tying this section directly to the pseudospectral analysis later.
Eigenvalue and eigenvector condition numbers
The central technical device is an auxiliary matrix $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$6 whose entries are $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$7, built from any basis $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$8 of the left null space of $\eig = (\eigcomp[1], \ldots, \eigcomp[m])$9. A genericity lemma shows that if the eigenvalue is algebraically simple, $\rvec$0 is nonsingular regardless of basis choice; the proof reduces a more general result for polynomial rMEPs to the linear case by selecting row-selection matrices yielding a coupled square problem and invoking Košir's lemma on its left eigenvector structure.
With this, the relative eigenvalue condition number is
$\rvec$1
and the bound is sharp: explicit rank-one perturbations attain it. For $\rvec$2 this recovers the classical GEP expression $\rvec$3. The relative form is invalid for zero eigenvalues, where the absolute variant must be used — a stated limitation. Since all ingredients depend continuously on the evaluated eigenvalue, computing the condition number at a computed approximation $\rvec$4 is justified whenever the backward error is small.
A notable structural result connects conditioning to geometry: for two-parameter problems with $\rvec$5 coefficient matrices, the Jacobian of the secular system equals a fixed diagonal rescaling times $\rvec$6. Consequently, tangential intersections of the secular curves correspond to large condition numbers — nearly parallel curves mean a nearly singular auxiliary matrix. Numerical examples bear this out quantitatively: in one example, condition numbers of 37.9, 62.9, and 12.0 correlate with mean intersection angles of 0.098, 0.208, and 0.628 radians respectively.
The eigenvector condition number, defined under a linear normalization $\rvec$7, is
$\rvec$8
with $\rvec$9 spanning the normalization complement and $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$0 the orthogonal complement of $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$1. Nonsingularity of the projected matrix $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$2 follows from a Schur-complement argument using regularity of the pencil away from the spectrum.
Multiparameter pseudospectrum
The $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$3-pseudospectrum collects all tuples admitting an exact solution of some perturbed problem with $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$4. Three equivalent characterizations are proven: existence of a unit vector with $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$5; the smallest-singular-value test $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$6; and the resolvent-type bound $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$7. The second characterization makes computation straightforward: evaluate $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$8 over a grid in $\mep{\eig}\rvec = \left(\mat_0 + \sum_{i=1}^m \eigcomp[i]\mat_i\right)\rvec = 0.$9.
Two structural lemmas relate the rMEP spectrum and pseudospectrum to those of square submatrix polynomials $\mep{\eig}$0:
- The spectrum equals the intersection over all $\mep{\eig}$1 submatrix spectra.
- The pseudospectrum satisfies only the inclusion $\mep{\eig}$2; the converse fails because deleting rows cannot increase $\mep{\eig}$3.
For visualization, the paper introduces (nested) right definite rMEPs, for which all eigenvalues are real and two-parameter pseudospectra can be drawn in the plane. However, the authors concede this class is very restrictive: no general construction is known for $\mep{\eig}$4 or $\mep{\eig}$5, and the proof is non-constructive; Hankel-structured $\mep{\eig}$6 problems provide the main known source of examples.
Computationally, since no simultaneous reduction of $\mep{\eig}$7 rectangular matrices exists, the pseudospectrum is computed slice-wise: fix $\mep{\eig}$8 parameters, treat the slice as a one-parameter rectangular pencil, and apply trapezoidal-form preprocessing (QZ for the slightly tall case $\mep{\eig}$9; QR for the very tall case L=(ℓk)0), followed by banded QR and inverse Lanczos iteration per grid point. Total complexity scales as L=(ℓk)1 up to preprocessing terms. MATLAB implementations (mperr, mpcond, mppseudo) accompany the paper.
Application to system identification
In least squares realization problems, the first-order optimality conditions form a multivariate polynomial system whose roots are eigenvalues of an associated rMEP; the global optimum is the eigenvalue minimizing the misfit. On a two-parameter example with data-generating model L=(ℓk)2, the real stationary points exhibit a striking pattern:
| Stationary point |
L=(ℓk)3 |
Condition number |
Cost |
Type |
| L=(ℓk)4 |
L=(ℓk)5 |
L=(ℓk)6 |
0 |
minimum |
| L=(ℓk)7 |
L=(ℓk)8 |
L=(ℓk)9 |
51.81 |
maximum |
| ℓ0 |
ℓ1 |
ℓ2 |
51.81 |
maximum |
| ℓ3 |
ℓ4 |
ℓ5 |
45.43 |
saddle |
| ℓ6 |
ℓ7 |
ℓ8 |
1.64 |
saddle |
The saddle point far from the data-generating parameters has a condition number roughly four orders of magnitude larger than the others, and its pseudospectral pocket is visibly stretched — consistent with tangential secular intersections. Motivated by this and related experiences in autoregressive moving-average identification, the paper states, explicitly without proof, a conjecture: in optimization-driven identification problems admitting multiparameter reformulations, the global optima are among the best-conditioned eigenvalues of the associated rMEP. If true, this is favorable for numerical practice, since the solutions of interest would be precisely the well-conditioned ones. The evidence remains empirical and limited to small examples.
Limitations and open questions
Several restrictions are acknowledged in the paper itself. The analysis covers only linear problems with isolated, finite, simple eigenvalues, although most results are expected to extend to polynomial rMEPs. Only norm-wise perturbations are treated; element-wise extensions analogous to structured GEP results are anticipated but not carried out. The relative eigenvalue condition number breaks down at zero eigenvalues. Right definiteness, needed for planar pseudospectrum visualization, lacks general constructions beyond Hankel-type ℓ9 cases. The optimality–conditioning conjecture is unproven. Open directions named by the authors include mixed subordinate and chordal norms, spherical projections of the pseudospectrum, and certification techniques for multiparameter eigenvalues.
Conclusion
The paper supplies the first systematic perturbation framework for rectangular multiparameter eigenvalue problems: computable norm-wise backward errors for eigenpairs and eigenvalues, sharp condition numbers for eigenvalues and eigenvectors built on a nonsingularity lemma for the auxiliary matrix, equivalent characterizations and slice-wise algorithms for the multiparameter pseudospectrum, and a geometric interpretation linking secular-curve intersection angles to conditioning. Its most consequential claim — that globally optimal solutions of multiparameter-reformulable optimization problems tend to be best-conditioned — is supported numerically but remains conjectural, and establishing it rigorously is the clearest open question raised by this work.