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On Hankel matrices and the symmetric nonnegative inverse eigenvalue problem.
- Source :
- Linear & Multilinear Algebra; Feb2024, Vol. 72 Issue 3, p474-487, 14p
- Publication Year :
- 2024
-
Abstract
- An n-by-n real symmetric matrix $ H=[h_{i,j}] $ H = [ h i , j ] is said to be a Hankel matrix if $ h_{i,j}=h_{i-1,j+1} $ h i , j = h i − 1 , j + 1 , for each $ i=2,\ldots,n $ i = 2 , ... , n and $ j=1,\ldots,n-1 $ j = 1 , ... , n − 1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if $ \sum _{i=1}^{n}\lambda _{i}=0 $ ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]
- Subjects :
- SYMMETRIC matrices
INVERSE problems
NONNEGATIVE matrices
REAL numbers
Subjects
Details
- Language :
- English
- ISSN :
- 03081087
- Volume :
- 72
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Linear & Multilinear Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 175362450
- Full Text :
- https://doi.org/10.1080/03081087.2022.2159306