[Paper Review] An informal introduction to perturbations of matrices determined up to similarity or congruence

Lena Klimenko, Vladimir V. Sergeichuk|arXiv (Cornell University)|Nov 4, 2013

Matrix Theory and Algorithms10 references5 citations

TL;DR

This paper provides an informal yet rigorous survey of perturbation theory for complex matrices under similarity, congruence, and *congruence transformations. It introduces miniversal deformations—normal forms that depend continuously on perturbations and capture all nearby matrices up to the given equivalence—offering a stable alternative to discontinuous canonical forms like the Jordan form.

ABSTRACT

The reductions of a square complex matrix A to its canonical forms under transformations of similarity, congruence, or *congruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the entries of A. We survey results about their behavior under perturbations of A and about normal forms of all matrices A+E in a neighborhood of A with respect to similarity, congruence, or *congruence. These normal forms are called miniversal deformations of A; they are not uniquely determined by A+E, but they are simple and depend continuously on the entries of E.

Motivation & Objective

• To address the instability of canonical forms (e.g., Jordan form) under small matrix perturbations, which depend discontinuously on entries.
• To introduce miniversal deformations as a continuous, stable alternative to canonical forms under similarity, congruence, and *congruence.
• To characterize the closure relations between matrix similarity, congruence, and *congruence classes using Hasse diagrams (closure graphs).
• To provide constructive methods for computing miniversal deformations and analyzing their structure via block matrices with independent parameters.
• To extend Arnold’s theory of miniversal deformations from similarity to congruence and *congruence, including explicit classification of canonical forms and closure relations.

Proposed method

• Constructs miniversal deformations for Jordan matrices under similarity by adding minimal independent parameters (stars) to off-diagonal blocks in a block-diagonal structure.
• Uses analytic transformations $\mathcal{S}(X)$ that depend smoothly on perturbation $X$ to reduce $J+X$ to the miniversal form $J+\mathcal{D}$, ensuring continuity.
• Applies the concept of codimension to determine the minimal number of independent parameters in the miniversal deformation.
• Extends the framework to congruence and *congruence by classifying canonical matrices using blocks of the form $\begin{bmatrix} \lambda & 0 \\ 0 & 0 \end{bmatrix}$ and $\begin{bmatrix} 0 & \tau \\ \tau & i\tau \end{bmatrix}$, with real nonnegative parameters.
• Constructs closure graphs (Hasse diagrams) showing inclusion relations between similarity and *congruence classes, where directed paths indicate closure relations.
• Uses geometric and algebraic criteria (e.g., nonnegative imaginary parts of inner products) to determine existence of arrows between canonical forms in the closure graph.

Experimental results

Research questions

• RQ1 How can one construct a normal form for matrices near a given matrix that depends continuously on perturbations, avoiding the discontinuities of the Jordan form?
• RQ2 What is the minimal number of independent parameters needed to represent all matrices close to a given matrix under similarity, congruence, or *congruence?
• RQ3 Which matrix classes can be approached arbitrarily closely by perturbations of a given matrix under similarity, congruence, or *congruence?
• RQ4 How can closure relations between matrix equivalence classes be systematically represented and characterized?
• RQ5 What are the structural and parametric conditions under which one canonical matrix is in the closure of another under *congruence?

Key findings

• Miniversal deformations under similarity are constructed as $J+\mathcal{D}$, where stars in off-diagonal blocks are replaced by independent complex parameters, and the number of such parameters equals the codimension of the similarity class.
• For a Jordan matrix $J$, all sufficiently small perturbations $J+X$ can be reduced via analytic similarity transformations to the miniversal form $J+\mathcal{D}$, ensuring continuous dependence on $X$.
• The closure graph for similarity classes of $n\times n$ matrices is a Hasse diagram showing that $J' \preccurlyeq J$ if $J'$ lies in the closure of the similarity class of $J$, with examples showing transitions like $J_2(\lambda)\oplus J_2(\lambda) \to J_3(\lambda)\oplus J_1(\lambda)$ or $J_4(\lambda)$ under small perturbations.
• Under *congruence, canonical forms are direct sums of blocks $\begin{bmatrix} \lambda & 0 \\ 0 & 0 \end{bmatrix}$ and $\begin{bmatrix} 0 & \tau \\ \tau & i\tau \end{bmatrix}$, with closure relations determined by real nonnegative combinations and imaginary part conditions.
• The closure of a *congruence class of a canonical matrix $M$ is the union of all *congruence classes of matrices $N$ for which there is a directed path from $N$ to $M$ in the closure graph.
• The closure graph for $2\times 2$ *congruence classes is infinite, with each vertex (except zero) representing an uncountable family of matrices parameterized by complex or real parameters, and arrows exist based on algebraic conditions on parameters (e.g., $\operatorname{Im}(\lambda \bar{\tau}) \geq 0$).