On the smallest singular value in the class of unit lower triangular matrices with entries in [-a, a]

ALTINIŞIK E.

SPECIAL MATRICES, cilt.9, sa.1, ss.297-304, 2021 (ESCI) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 9 Sayı: 1
  • Basım Tarihi: 2021
  • Doi Numarası: 10.1515/spma-2020-0139
  • Dergi Adı: SPECIAL MATRICES
  • Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus
  • Sayfa Sayıları: ss.297-304
  • Anahtar Kelimeler: real symmetric matrix, unit lower triangular matrix, GCD and LCM matrix, smallest eigenvalue, smallest singular value, LARGEST EIGENVALUES, CONJECTURE, BEHAVIOR, GCD
  • Gazi Üniversitesi Adresli: Evet

Özet

Given a real number a >= 1, let K-n(a) be the set of all n x n unit lower triangular matrices with each element in the interval [-a, a]. Denoting by lambda(n)(center dot) the smallest eigenvalue of a given matrix, let c(n)(a) = min {lambda(n)(YYT) : Y epsilon K-n(a)}. Then root c(n)(a) is the smallest singular value in K-n(a). We find all minimizing matrices. Moreover, we study the asymptotic behavior of c(n)(a) as n -> infinity. Finally, replacing [-a, a] with [a, b], a <= 0 < b, we present an open question: Can our results be generalized in this extension?