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Estimates of the determinant of a perturbed identity matrix
Publikationstyp
Journal Article
Date Issued
2018-12-01
Sprache
English
Author(s)
Institut
TORE-URI
Volume
558
Start Page
101
End Page
107
Citation
Linear Algebra and Its Applications (558): 101-107 (2018-12-01)
Publisher DOI
Scopus ID
Recently Brent et al. presented new estimates for the determinant of a real perturbation I+E of the identity matrix. They give a lower and an upper bound depending on the maximum absolute value of the diagonal and the off-diagonal elements of E, and show that either bound is sharp. Their bounds will always include 1, and the difference of the bounds is at least tr(E). In this note we present a lower and an upper bound depending on the trace and Frobenius norm ϵ:=‖E‖Fof the (real or complex) perturbation E, where the difference of the bounds is not larger than ϵ2+O(ϵ3) provided that ϵ<1. Moreover, we prove a bound on the relative error between det(I+E) and exp(tr(E)) of order ϵ2.