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Generic real Jordan canonical forms

Fernando De Terán, Froilán M. Dopico

TL;DR

This work determines the generic real Jordan canonical forms (RJCF) of real $n\times n$ matrices under real similarity by introducing and classifying real similarity bundles. It proves that $\mathbb{R}^{n\times n}$ is the union of closures of $\lfloor n/2\rfloor+1$ open, disjoint generic bundles, each specified by the number $t$ of nonreal eigenvalue pairs and containing matrices with $t$ distinct nonreal conjugate pairs and $n-2t$ distinct real eigenvalues; these bundles are described via blocks $C(a_i,b_i)$ for complex-conjugate pairs and real blocks $[c_i]$, ensuring eigenvalue distinctness within a bundle. A key lemma connects perturbations and real Schur form to show that the bundles have codimension $0$ and that their closures do not intersect, yielding an open dense decomposition of the matrix space. The paper also presents numerical experiments and connects the results to random-matrix theory (via probabilities $p_{n,k}$ for the number of real eigenvalues), confirming the generic bundles’ appearance and alignment with theoretical predictions. Overall, the work provides a complete topological and constructive classification of generic RJCFs for real matrices and clarifies the typical real-eigenvalue distributions in this setting.

Abstract

We obtain the generic real Jordan canonical forms for $n\times n$ matrices with real entries. More precisely, we prove that the set of $n\times n$ real matrices is the union of the closures of $\lfloor n/2\rfloor+1$ sets, which are called generic bundles, as they are particular "bundles". In general, a bundle is the set of $n\times n$ real matrices with the same real Jordan canonical form, up to the values of the eigenvalues, provided that the eigenvalues which are distinct in one matrix of the bundle remain distinct in any other matrix of the same bundle. The $k$th generic bundle, for $0\leq k\leq\lfloor n/2\rfloor$, contains the $n\times n$ real matrices having $k$ different pairs of non-real conjugate eigenvalues and $n-2k$ different real eigenvalues. We prove that each of the $\lfloor n/2\rfloor+1$ generic bundles is an open subset of the set of $n\times n$ real matrices. Some numerical experiments are carried out with large sets of random matrices of different sizes to confirm that all the generic bundles show up, and only these ones.

Generic real Jordan canonical forms

TL;DR

This work determines the generic real Jordan canonical forms (RJCF) of real matrices under real similarity by introducing and classifying real similarity bundles. It proves that is the union of closures of open, disjoint generic bundles, each specified by the number of nonreal eigenvalue pairs and containing matrices with distinct nonreal conjugate pairs and distinct real eigenvalues; these bundles are described via blocks for complex-conjugate pairs and real blocks , ensuring eigenvalue distinctness within a bundle. A key lemma connects perturbations and real Schur form to show that the bundles have codimension and that their closures do not intersect, yielding an open dense decomposition of the matrix space. The paper also presents numerical experiments and connects the results to random-matrix theory (via probabilities for the number of real eigenvalues), confirming the generic bundles’ appearance and alignment with theoretical predictions. Overall, the work provides a complete topological and constructive classification of generic RJCFs for real matrices and clarifies the typical real-eigenvalue distributions in this setting.

Abstract

We obtain the generic real Jordan canonical forms for matrices with real entries. More precisely, we prove that the set of real matrices is the union of the closures of sets, which are called generic bundles, as they are particular "bundles". In general, a bundle is the set of real matrices with the same real Jordan canonical form, up to the values of the eigenvalues, provided that the eigenvalues which are distinct in one matrix of the bundle remain distinct in any other matrix of the same bundle. The th generic bundle, for , contains the real matrices having different pairs of non-real conjugate eigenvalues and different real eigenvalues. We prove that each of the generic bundles is an open subset of the set of real matrices. Some numerical experiments are carried out with large sets of random matrices of different sizes to confirm that all the generic bundles show up, and only these ones.
Paper Structure (4 sections, 2 theorems, 12 equations, 3 tables)

This paper contains 4 sections, 2 theorems, 12 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and basic definitions
  3. The main result
  4. Real eigenvalues of random matrices: Numerical experiments

Key Result

Lemma 3.1

If $M\in{\overline{\mathcal{B}}}_{\mathbb R} \left(\bigoplus_{i=1}^tC(a_i,b_i)\oplus\bigoplus_{i=1}^{n-2t}[c_i]\right)$, for some $0\leq t\leq\lfloor n/2\rfloor$, with $a_i, b_i, c_i\in\mathbb R$, $b_i >0$, then

Theorems & Definitions (4)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof