Inverse Eigenvalue Problems in Matrix Theory

Inverse eigenvalue problems concern the reconstruction of a matrix from prescribed spectral characteristics. Central to this area is the question of existence and uniqueness: given a set of eigenvalues and, where relevant, partial eigenvectors or extremal spectral data of leading principal submatrices, can one determine a matrix within a certain structural class? Considerable progress has been made for classes of symmetric and non-symmetric matrices, including banded forms such as tridiagonal or pentadiagonal matrices, and matrices whose adjacency patterns correspond to graphs or trees. Techniques draw on interlacing inequalities, recurrence relations among characteristic polynomials, combinatorial constructs associated with graph topology and perturbation results extended to non-Hermitian settings. Algorithmic schemes have been devised for explicit construction, often employing modified Lanczos procedures or combinatorial labelling of graph vertices. These problems have a global significance across disciplines: in vibration analysis of mechanical systems, signal processing, quantum mechanics and network science. Advances in theoretical characterisation now allow practitioners to tailor matrix structures to desired dynamic responses or connectivity constraints, while ensuring computational feasibility. Ongoing research explores the interplay between graph-theoretic constraints, genericity of eigenvectors and the exploitation of redundant spectral data to guarantee robust reconstruction.

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Recent work has generalised the classical Hermite–Biehler theorem to address direct and inverse spectral problems for non-Hermitian rank-one and rank-two perturbations of finite Hermitian and Jacobi matrices. This broadens the applicability of interlacing and zero-configuration results to a setting where one eigenvalue of the perturbation breaks strict interlacing, providing full characterisation of the resulting zero distribution and constructive methods for matrix recovery.

Advances in structured banded matrices are exemplified by the derivation of sufficient conditions for reconstructing symmetric and nonsymmetric pentadiagonal matrices from combinations of extremal eigenvalues of leading principal minors, selected eigenvectors and prescribed entries. The constructive approach utilises modified Lanczos algorithms and interlacing criteria to yield families of solutions, demonstrating flexible control over matrix entries while maintaining specified spectral patterns.

In the graph-theoretic realm, the inverse problem has been completely solved for clique-path and related block graphs by exploiting a strong spectral property. A novel technique shows how appending a clique to a graph allows the insertion of arbitrary eigenvalues into the spectrum, establishing the realisability of diverse spectral lists and expanding the catalogue of adjacency patterns compatible with prescribed eigendata.

Inverse Eigenvalue Problems in Matrix Theory publication trend

The graph below shows the total number of articles in inverse eigenvalue problems in matrix theory across all publications each year (not limited to Nature Index journals).

Technical terms

Inverse eigenvalue problem: The task of constructing a matrix with specified eigenvalues and, optionally, additional spectral or structural constraints.

Leading principal submatrix: A submatrix obtained by deleting the last n–k rows and columns of an n×n matrix, used to impose nested spectral data.

Interlacing: A relation between spectra of nested submatrices wherein eigenvalues of the submatrix lie between those of the parent matrix.

Jacobi matrix: A symmetric tridiagonal matrix often arising in orthogonal polynomial theory and spectral discretisation.

Pentadiagonal matrix: A banded matrix with nonzero entries on the main diagonal and the two adjacent upper and lower diagonals, notable for sparse representation.

Graph of a matrix: A representation in which nonzero off-diagonal entries correspond to edges in an associated graph, linking spectral properties to topology.

References

1. A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices. Journal of Mathematical Analysis and Applications (2024).
2. Two Inverse Eigenproblems for Certain Symmetric and Nonsymmetric Pentadiagonal Matrices. Mathematics (2022).
3. On the inverse eigenvalue problem for block graphs. Linear Algebra and its Applications (2021).
4. Distinct eigenvalues are realizable with generic eigenvectors. Linear and Multilinear Algebra (2023).