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Takagi factorization of matrices depending on parameters and locating degeneracies of singular values

Luca Dieci, Alessandra Papini, Alessandro Pugliese

TL;DR

This paper investigates Takagi factorization for matrix-valued functions dependent on parameters, focusing on smoothness of the Takagi factors and degeneracies where singular values coalesce or vanish. It proves that double or zero singular values are real codimension $2$ phenomena for complex symmetric matrices and connects Takagi factors to a symmetric eigenproblem to analyze conical intersections. A differential-equation framework and a predictor-corrector algorithm are developed to compute smooth Takagi factorizations along one-parameter paths and around loops in parameter space. Numerical experiments on two-parameter random ensembles reveal quadratic scaling for coalescences and linear scaling for rank losses with matrix size, and they propose a density model for degeneracies based on the quarter-circle law. Overall, the work provides both theoretical insight and practical tools for locating and continuing degeneracies in parameter-dependent Takagi factorizations.

Abstract

In this work we consider the Takagi factorization of a matrix valued function depending on parameters. We give smoothness and genericity results and pay particular attention to the concerns caused by having either a singular value equal to $0$ or multiple singular values. For these phenomena, we give theoretical results showing that their co-dimension is $2$, and we further develop and test numerical methods to locate in parameter space values where these occurrences take place. Numerical study of the density of these occurrences is performed.

Takagi factorization of matrices depending on parameters and locating degeneracies of singular values

TL;DR

This paper investigates Takagi factorization for matrix-valued functions dependent on parameters, focusing on smoothness of the Takagi factors and degeneracies where singular values coalesce or vanish. It proves that double or zero singular values are real codimension phenomena for complex symmetric matrices and connects Takagi factors to a symmetric eigenproblem to analyze conical intersections. A differential-equation framework and a predictor-corrector algorithm are developed to compute smooth Takagi factorizations along one-parameter paths and around loops in parameter space. Numerical experiments on two-parameter random ensembles reveal quadratic scaling for coalescences and linear scaling for rank losses with matrix size, and they propose a density model for degeneracies based on the quarter-circle law. Overall, the work provides both theoretical insight and practical tools for locating and continuing degeneracies in parameter-dependent Takagi factorizations.

Abstract

In this work we consider the Takagi factorization of a matrix valued function depending on parameters. We give smoothness and genericity results and pay particular attention to the concerns caused by having either a singular value equal to or multiple singular values. For these phenomena, we give theoretical results showing that their co-dimension is , and we further develop and test numerical methods to locate in parameter space values where these occurrences take place. Numerical study of the density of these occurrences is performed.

Paper Structure

This paper contains 7 sections, 7 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction. Takagi factorization of a matrix.
  2. Smoothness results, degenerate singular values, codimension.
  3. One parameter case.
  4. Differential equations for the factors.
  5. A predictor corrector algorithm to compute the factorization along loops in parameter space.
  6. Numerical tests.
  7. Conclusions

Key Result

Lemma 1.1

$\,$ Existence. Let $A\in {{\mathbb C}^{n\times n}}$ be symmetric: $A=A^T$. Then, $A$ admits a Takagi factorization $A=USU^T$, where $U$ is unitary and $S$ is the diagonal matrix of singular values of $A$, which we will take to be ordered: $S=\operatorname{diag}(\sigma_1,\dots, \sigma_n), \,\ \sigma

Figures (1)

  • Figure 1: The figure shows the results of a "log--log least squares" performed on the data obtained from our experiments (circles for number of coalescences of singular values, diamonds for losses of rank). It also shows the parameters $c$ and $q$ obtained by best fit for the power law: $\text{ \# of points } = c\,n^q, \text{ where } n \text{ is the matrix dimension.}$

Theorems & Definitions (22)

  • Lemma 1.1: Takagi Factorization
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • ...and 12 more