PreprintCC BY 4.0arXiv
Isospectral Deformations of Diagonal Matrices by Symmetric Matrices in Dimension Four
Abstract
For a diagonal matrix D with distinct diagonal entries in an algebraically closed field K we study the symmetric matrices Q with entries in K such that the affine line D+tQ has the same eigenvalues in K as D. We study the equality of characteristic polynomials, χD+tQ(x)=χD(x) for all t∈ K and examine the resulting system of equations. For generic diagonal matrices of size at most $4$ we show that only the trivial symmetric matrix Q=0 satisfies this isospectral condition. We use this result to compute the degree of a field extension that appears in the invariant theory of quadratic forms under congruence action.
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BibTeX
@misc{asokan2026isospectral,
title = {{Isospectral Deformations of Diagonal Matrices by Symmetric Matrices in Dimension Four}},
author = {Hari Asokan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07150},
eprint = {2611.07150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07150},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Isospectral Deformations of Diagonal Matrices by Symmetric Matrices in Dimension Four
AU - Hari Asokan
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07150
UR - https://arxiv.org/abs/2611.07150
AB - For a diagonal matrix D with distinct diagonal entries in an algebraically closed field K we study the symmetric matrices Q with entries in K such that the affine line D+tQ has the same eigenvalues in K as D. We study the equality of characteristic polynomials, χ_D+tQ(x)=χ_D(x) for all t∈ K and examine the resulting system of equations. For generic diagonal matrices of size at most $4$ we show that only the trivial symmetric matrix Q=0 satisfies this isospectral condition. We use this result to compute the degree of a field extension that appears in the invariant theory of quadratic forms under congruence action.
ER -