Generic real Jordan canonical forms
Fernando De Ter\'an, Froil\'an M. Dopico

TL;DR
This paper characterizes the generic real Jordan canonical forms for real matrices, showing they form a union of open sets called generic bundles distinguished by the number of real and complex conjugate eigenvalue pairs.
Contribution
It introduces the concept of generic bundles for real Jordan forms and proves these bundles are open and cover the entire space of real matrices.
Findings
The set of real matrices is the union of closures of generic bundles.
Each generic bundle corresponds to matrices with a fixed number of real and complex conjugate eigenvalues.
Numerical experiments confirm all generic bundles are represented in random matrices.
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…
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Taxonomy
TopicsMatrix Theory and Algorithms · Random Matrices and Applications · Tensor decomposition and applications
