Nonlinear Kalman Varieties Advance Understanding of Matrices with Structured Eigenvectors

The study of matrices with eigenvectors constrained to lie on specific algebraic varieties presents a fascinating challenge in mathematics, with growing relevance to fields like optimisation. Flavio Salizzoni, Luca Sodomaco, and Julian Weigert investigate these so-called nonlinear Kalman varieties, extending earlier work on linear versions first explored by Ottaviani and Sturmfels. This research significantly advances understanding of the fundamental properties of these varieties, including their dimensions, degrees, and the nature of their singularities. Importantly, the team provides a novel determinantal description for the equations defining Kalman varieties associated with hypersurfaces, offering a powerful new tool for analysis and computation in this area.

Importantly, the team provides a novel determinantal description for the equations defining Kalman varieties associated with hypersurfaces, offering a powerful new tool for analysis and computation in this area.

Kalman varieties are algebraic varieties arising in control theory, optimization, and numerical linear algebra. This paper extends the theory to nonlinear settings, focusing on the geometry of these nonlinear varieties and their degrees, which measure their complexity. The degree of a variety is a fundamental invariant, indicating how complicated it is and how difficult it may be to analyse.

The research centres on defining and characterizing Kalman varieties, representing the set of matrices satisfying a specific eigenvector condition. Scientists developed a framework for studying these varieties when eigenvectors lie on nonlinear algebraic varieties, a significant generalization of existing theory. They derive formulas for the degrees of certain nonlinear Kalman varieties, providing valuable insights into their geometry and establishing connections to optimization problems, such as optimizing the Rayleigh quotient.

Kalman Varieties and Eigenvector Constraints

Scientists developed a novel approach to study the locus of square matrices possessing at least one eigenvector lying on a prescribed algebraic variety, extending previous work to encompass nonlinear varieties. This research centers on defining and characterizing “Kalman varieties,” determining their fundamental geometric properties, including dimension, degree, and singularities. The study employs determinantal equations, generalizing techniques established for linear cases to the more complex setting of hypersurfaces.

To achieve this, researchers constructed a system of polynomial equations representing the conditions for eigenvector alignment on the given variety. They leveraged a classical technique involving the computation of Jacobian polynomials to establish relationships between the matrix elements and the defining equations of the variety. Specifically, the method involves solving a system of equations and identifying a resultant polynomial, which serves as the defining equation for the Kalman variety. For a specific example involving a conic section, the team applied this method, constructing a system of quadratic equations and computing the resultant using specialized computational tools. Verification was achieved through computational methods, demonstrating that the resultant could be expressed as a product of two polynomials, providing a clear representation of the variety’s equation. This innovative approach extends beyond conics, offering a framework for analysing Kalman varieties associated with a broader range of algebraic varieties and matrix types.

Nonlinear Kalman Varieties and Symmetric Powers

Researchers investigated Kalman varieties when defined over an arbitrary projective variety, extending previous work focused solely on linear subspaces. The study establishes fundamental properties of these nonlinear Kalman varieties, including their irreducibility, codimension, and degree, providing a foundational understanding of their geometric characteristics.

A key achievement is the development of a new matrix, termed the Kalman matrix of order d, constructed by replacing the original matrix A with its dth symmetric power. This new matrix plays a crucial role in describing the equations that define the nonlinear Kalman variety. Specifically, the researchers demonstrate that the vanishing of maximal minors of this new Kalman matrix contains, but does not always perfectly define, the nonlinear Kalman variety. For a hypersurface cut out by a single homogeneous polynomial of degree d, the determinant of the nonlinear Kalman matrix exhibits several factors, and the team explicitly describes these factors and their multiplicities. Furthermore, the research delves into the singularities of nonlinear Kalman varieties, which are particularly challenging to study. By degenerating the variety to a union of hyperplanes, the team provides geometric intuition and determines the codimension and degree of the reduced singular locus of the Kalman variety.

Nonlinear Kalman Varieties and Determinantal Equations

This work presents a detailed investigation into Kalman varieties, which are algebraic varieties defined by matrices possessing at least one eigenvector on a prescribed variety. Building upon earlier research concerning linear Kalman varieties, the team extended the study to encompass the more general case of nonlinear varieties, specifically those associated with hypersurfaces. A key achievement lies in the derivation of determinantal-like equations that describe these nonlinear Kalman varieties, offering a new means of characterizing their geometric properties.

The research successfully determines fundamental invariants of these varieties, including their dimensions, degrees, and singular characteristics. These findings contribute to a deeper understanding of the algebraic structures underlying problems in optimisation and related fields. The team has made all computational examples and supporting software publicly available, facilitating further investigation and analysis by other researchers.