An Upper Bound for the Largest Eigenvalue of a Positive Semidefinite Block Banded Matrix
- Authors: Kolotilina L.Y.1
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Affiliations:
- St.Petersburg Department of the Steklov Mathematical Institute
- Issue: Vol 232, No 6 (2018)
- Pages: 917-920
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/241484
- DOI: https://doi.org/10.1007/s10958-018-3918-6
- ID: 241484
Cite item
Abstract
The new upper bound
\( {\uplambda}_{\mathrm{max}}(A)\le \sum \limits_{k=1}^{p+1}i\equiv {k}_{\left(\operatorname{mod}p+1\right)}^{\mathrm{max}}{\uplambda}_{\mathrm{max}}\left({A}_{ii}\right) \)
for the largest eigenvalue of a Hermitian positive semidefinite block banded matrix A = (Aij ) of block semibandwidth p is suggested. In the special case where the diagonal blocks of A are identity matrices, the latter bound reduces to the bound
\( {\uplambda}_{\mathrm{max}}(A)\le p+1, \)
depending on p only, which improves the bounds established for such matrices earlier and extends the bound
\( {\uplambda}_{\mathrm{max}}(A)\le 2, \)
old known for p = 1, i.e., for block tridiagonal matrices, to the general case p ≥ 1.
About the authors
L. Yu. Kolotilina
St.Petersburg Department of the Steklov Mathematical Institute
Author for correspondence.
Email: lilikona@mail.ru
Russian Federation, St.Petersburg