Matrix inversion cases with size-independent tensor rank estimates

Ivan Oseledets, Eugene Tyrtyshnikov, Nickolai Zamarashkin

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

Let M be a matrix of order n = pq. Then the tensor rank of M is defined as the minimal possible ρ in expressions of the form M = ∑t = 1ρ Ut ⊗ Vt, where Ut and Vt are matrices of order p and q, respectively. Let M be a nonsingular matrix of tensor rank 3 and, moreover, of the formM = I + A ⊗ X + Y ⊗ Bwith rank X = rank Y = 1. Then, it is discovered and proved that the tensor rank of M- 1 is bounded from above by 5 independently of p and q, the estimate being sharp. Some related and extended results are also given.

Original languageEnglish
Pages (from-to)558-570
Number of pages13
JournalLinear Algebra and Its Applications
Volume431
Issue number5-7
DOIs
Publication statusPublished - 1 Aug 2009
Externally publishedYes

Keywords

  • Circulant matrices
  • Inverse matrices
  • Kronecker product
  • Low-rank matrices
  • Multilevel matrices
  • Tensor ranks
  • Toeplitz matrices

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