Abstract
Abstract A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ. Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative matrix.
Highlights
The nonnegative inverse eigenvalue problem (NIEP) is the problem of characterizing all possible spectra of entrywise nonnegative matrices
A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix
In this paper we provide a new su cient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ
Summary
The nonnegative inverse eigenvalue problem (NIEP) is the problem of characterizing all possible spectra of entrywise nonnegative matrices. In [15], Paparella proved that the permutative RNIEP has a solution, when the given spectrum is of real Suleımanova type [24]. By applying Brauer’s Theorem, a very simple and short proof, that real Suleımanova spectra are permutatively realizable, was given in [23]. It was showed in [23], that a complex Suleımanova spectrum is in particular permutatively realizable. Loewy [12] gave a negative answer to the following question set by Paparella in [15]: can all realizable spectra of real numbers, be realized by a permutative matrix, or by a direct sum of permutative matrices?
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