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Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

Luca Dieci, Alessandra Papini, Alessandro Pugliese

TL;DR

The paper addresses the parametric generalized eigenproblem [A(x) - λ B(x)] v = 0 with symmetric $A(x)$ and positive-definite $B(x)$ depending on two parameters, focusing on when eigenvalues/eigenvectors remain smooth and how conical intersections (CI) arise as generic, codimension-2 phenomena. It develops a comprehensive smoothness theory (square roots, Cholesky factors, block-diagonalization, and differential equations for eigenpairs), extends to periodic pencils, and introduces algorithms based on predictor-corrector continuation to locate CIs, including strategies to handle veering via 2×2 blocks. A stochastic numerical study using SG$^+$ ensembles across bandwidths shows a power-law growth of conical intersections with problem dimension and demonstrates bandwidth as a key driver of CI counts, with dispersion having a modest effect. The results inform reduced-order projections and stability analyses in engineering contexts and provide practical tools for CI detection and quantification in parametric eigenvalue problems.

Abstract

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-λB(x)$, where $A$ and $B$ are symmetric matrix valued functions in ${\mathbb R}^{n\times n}$, smoothly depending on parameters $x\in Ω\subset {\mathbb R}^2$; further, $B$ is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where $A$ and $B$ are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.

Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

TL;DR

The paper addresses the parametric generalized eigenproblem [A(x) - λ B(x)] v = 0 with symmetric and positive-definite depending on two parameters, focusing on when eigenvalues/eigenvectors remain smooth and how conical intersections (CI) arise as generic, codimension-2 phenomena. It develops a comprehensive smoothness theory (square roots, Cholesky factors, block-diagonalization, and differential equations for eigenpairs), extends to periodic pencils, and introduces algorithms based on predictor-corrector continuation to locate CIs, including strategies to handle veering via 2×2 blocks. A stochastic numerical study using SG ensembles across bandwidths shows a power-law growth of conical intersections with problem dimension and demonstrates bandwidth as a key driver of CI counts, with dispersion having a modest effect. The results inform reduced-order projections and stability analyses in engineering contexts and provide practical tools for CI detection and quantification in parametric eigenvalue problems.

Abstract

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem , where and are symmetric matrix valued functions in , smoothly depending on parameters ; further, is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where and are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.

Paper Structure

This paper contains 14 sections, 22 theorems, 71 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model problem and eigenvalues continuity
  3. A collection of smoothness results
  4. Square root
  5. General block diagonalization results
  6. One parameter case: smoothness
  7. Differential Equations for the factors
  8. Periodicity
  9. Coalescing eigenvalues of \ref{['GenEv']}
  10. One generic coalescing in $\Omega$
  11. Several generic coalescing points in $\Omega$
  12. Algorithms to locate coalescing eigenvalues
  13. Random matrix ensemble and experiments
  14. Conclusions

Key Result

Lemma 1.2

For StaticGenEv, the eigenvector matrix $V\in {{\mathbb R}^{n\times n}}$, $V=[v_1,\dots, v_n]$, can be chosen so to satisfy the relation If the eigenvalues $\lambda_i$'s are distinct, then, for a given ordering of the eigenvalues, the matrix $V$ in B-ortho is unique up to the sign of its columns.

Figures (4)

  • Figure 1: Transversal Intersection at $\xi_0$
  • Figure 2: Figure for proof of Theorem \ref{['PencilAny']}
  • Figure 3: For each value of $b=3, 4, 5, \mathrm{full}$ and $\delta=0.05, 0.25, 0.45, 0.65, 0.85$, we have performed a log-log linear least squares regression. The Figure shows the best fit lines for all combinations of $b$ and $\delta$, and also the average exponent (slope after the log-log transformation) $p$ for each value of $b$.
  • Figure 4: default

Theorems & Definitions (56)

  • Remark 1.1
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • proof
  • Definition 2.1: $B$-orthogonality
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 46 more