arXiv:2606.20432

Eigenvector Varieties

Sandra Di Rocco, Bernd Sturmfels, Svala Sverrisdóttir

math.AGmath.RAquant-phSubmitted Jun 18, 2026• Announced Jun 19, 2026

Abstract

Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.

Related papers

The Linear Span of Uniform Matrix Product States

Universit&, Claudia De Lazzari, #224 +4 more

The variety of uniform matrix product states arises both in algebraic geometry as a natural generalization of the Veronese variety, and in quantum many-body physics as a model for a translation-invariant system of sites placed on a ring. Using methods from linear algebra, representation theory, and invariant theory of matrices, we study the linear span of this variety.

arXiv:2204.10363Symmetry Integrability and Geometry Methods and ApplicationsDec 21, 20223 citations

Quantum Hamiltonians associated with finite-dimensional Lie algebras and factorized s-matrices

David V. Chudnovsky, Gregory V. Chudnovsky

Lecture notes in mathematicsJan 1, 19824 citations

Eigenvector Continuation with Subspace Learning

Dillon Frame, Rongzheng He, Ilse C. F. Ipsen +3 more

A common challenge faced in quantum physics is finding the extremal eigenvalues and eigenvectors of a Hamiltonian matrix in a vector space so large that linear algebra operations on general vectors are not possible. There are numerous efficient methods developed for this task, but they generally fail when some control parameter in the Hamiltonian matrix exceeds some threshold value. In this Letter we present a new technique called eigenvector continuation that can extend the reach of these methods. The key insight is that while an eigenvector resides in a linear space with enormous dimensions, the eigenvector trajectory generated by smooth changes of the Hamiltonian matrix is well approximated by a very low-dimensional manifold. We prove this statement using analytic function theory and propose an algorithm to solve for the extremal eigenvectors. We benchmark the method using several examples from quantum many-body theory.

arXiv:1711.07090Physical Review LettersJul 17, 2018140 citations

Bethe eigenvectors of higher transfer matrices

Evgeny Mukhin, V. Tarasov, A Varchenko

We consider the XXX -type and Gaudin quantum integrable models associated with the Lie algebra . The models are defined on a tensor product of irreducible -modules. For each model, there exist N one-parameter families of commuting operators on , called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.

arXiv:math/0605015Journal of Statistical Mechanics Theory and ExperimentAug 3, 200653 citations

Eigenvalues and eigenvectors of a matrix

Helmut Bez, Tony Croft

This chapter introduces the essential topics and to show how eigenvalues and eigenvectors are calculated. In quantum computation it is relevant to consider cases in which the eigenvalues are all real and the eigenvectors are orthogonal. In the study of quantum computation, the matrices with which we usually work have properties that enable us to be quite specific about the nature of the eigenvalues and eigenvectors. Post-measurement the system is in a state represented by an eigenvector or eigenvectors. In quantum computation it is relevant to consider cases in which the eigenvalues are all real and the eigenvectors are orthogonal.

Chapman & Hall/CRC eBooksJan 6, 20235 citations

Bethe Eigenvectors of Higher Transfer Matrices

Evgeny Mukhin

We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra gl N. The models are defined on a tensor product M1 ⊗... ⊗ Mn of irreducible gl N-modules. For each model, there exist N oneparameter families of commuting operators on M1 ⊗... ⊗ Mn, called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.

Jan 1, 200688 citations