On The Doubly Stochastic Realization Of Spectra (original) (raw)
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NONNEGATIVE REALIZATION OF COMPLEX SPECTRA
2010
We consider a list of complex numbers Λ = {λ 1 , λ 2 , . . . , λn} and give a simple and efficient sufficient codition for the existence of an n × n nonnegative matrix with spectrum Λ. Our result extends a previous one for a list of real numbers given in [LAA 416 (2006) 844-856]. In particular, we show how to construct a nonnegative matrix with prescribed complex eigenvalues and diagonal entries. As a by-product we also construct Hermitian matrices with prescribed spectrum, whose entries have nonnegative real parts.
On universal realizability of spectra
Linear Algebra and its Applications, 2018
A list Λ = {λ 1 , λ 2 ,. .. , λ n } of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be universally realizable (U R) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an n × n nonnegative matrix A is co-spectral to a nonnegative matrix B with constant row sums. In this paper, we extend the co-spectrality between A and B to a similarity between A and B, when the Perron eigenvalue is simple. We also show that if ǫ ≥ 0 and Λ = {λ 1 , λ 2 ,. .. , λ n } is U R, then {λ 1 + ǫ, λ 2 ,. .. , λ n } is also
A note on a relationship between
2012
An n-list λ := (r; λ 2 ,. .. , λ n) of complex numbers with r > 0, is said to be realizable if λ is the spectrum of n × n nonnegative matrix A and in this case A is said to be a nonnegative realization of λ. If, in addition, each row and column sum of A equals r, then λ is said to be doubly stochastically realizable and in such case A is said to be a doubly stochastic realization for λ. In 1997, Guo proved that if (λ 2 ,. .. , λ n) is any list of complex numbers which is closed under complex conjugation then there exists a least real number λ 0 with max 2≤j≤n |λ j | ≤ λ 0 ≤ 2n max 2≤j≤n |λ j | such that the list of complex numbers {λ 1 , λ 2 , ..., λ n } is realizable if and only if λ 1 ≥ λ 0. In 2020, Julio and Soto showed that the upper bound may be reduced to (n − 1) max 2≤j≤n |λ j | in the case when at least one of the λ i is real. In this paper, we first describe an algorithm for passing from a nonnegative realization to a doubly stochastic realization. As applications, we give a new sufficient condition for a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in this case B is shown to be the unique closest doubly stochastic matrix to A with respect to the Frobenius norm. Then, our next results slightly improve the upper bound for the nonnegative realization presented by Julio and Soto and in the case when none of the λ i is real, we also give an improvement of Guo's bound. Then, for doubly stochastic realizations, we obtain an upper bound that improves Guo's bound as well. Finally, for certain particular cases, we give a further improvement of our last bound for doubly stochastic realization.
On the symmetric doubly stochastic matrices that are determined by their spectra
arXiv (Cornell University), 2013
A symmetric doubly stochastic matrix A is said to be determined by its spectra if the only symmetric doubly stochastic matrices that are similar to A are of the form P T AP for some permutation matrix P. The problem of characterizing such matrices is considered here. An "almost" the same but a more difficult problem was proposed by [ M. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Lin. Alg. Appl., 432 (2010) 2925-2927] as follows: "Characterize all the n-tuples λ = (1, λ 2 , ..., λ n) such that up to a permutation similarity, there exists a unique symmetric doubly stochastic matrix with spectrum λ." In this short note, some general results concerning our two problems are first obtained. Then, we completely solve these two problems for the case n = 3. Some connections with spectral graph theory are then studied. Finally, concerning the general case, two open questions are posed and a conjecture is introduced.
On nonnegative realization of partitioned spectra
We consider partitioned lists of real numbers Λ = {λ 1 , λ 2 , . . . , λn}, and give efficient and constructive sufficient conditions for the existence of nonnegative and symmetric nonnegative matrices with spectrum Λ. Our results extend the ones given in [R.L. Soto and O. Rojo. Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl., 416:844-856, 2006.] and [R.L. Soto, O. Rojo, J. Moro, and A. Borobia. Symmetric nonnegative realization of spectra. Electron. J. Linear Algebra, 16:1 -18, 2007.] for the real and symmetric nonnegative inverse eigenvalue problem. We also consider the complex case and show how to construct an r × r nonnegative matrix with prescribed complex eigenvalues and diagonal entries.
Nonnegative realization of spectra having negative real parts
Linear Algebra and its Applications, 2006
The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list σ except for one (the Perron eigenvalue) have real parts smaller than or equal to zero.
Linear and Multilinear Algebra, 2014
A symmetric doubly stochastic matrix A is said to be determined by its spectra if the only symmetric doubly stochastic matrices that are similar to A are of the form P T AP for some permutation matrix P. The problem of characterizing such matrices is considered here. An "almost" the same but a more difficult problem was proposed by [ M. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Lin. Alg. Appl., 432 (2010) 2925-2927] as follows: "Characterize all the n-tuples λ = (1, λ 2 , ..., λ n) such that up to a permutation similarity, there exists a unique symmetric doubly stochastic matrix with spectrum λ." In this short note, some general results concerning our two problems are first obtained. Then, we completely solve these two problems for the case n = 3. Some connections with spectral graph theory are then studied. Finally, concerning the general case, two open questions are posed and a conjecture is introduced.
Existence and construction of nonnegative matrices with complex spectrum
Linear Algebra and its Applications, 2003
The following inverse spectrum problem for nonnegative matrices is considered: given a set of complex numbers σ = {λ 1 , λ 2 , . . . , λ n }, find necessary and sufficient conditions for the existence of an n × n nonnegative matrix A with spectrum σ . Our work is motivated by a relevant theoretical result of Guo Wuwen [Linear Algebra Appl. 266 (1997) 261, Theorem 2.1]: there exists a real parameter λ 0 max 2 j n |λ j | such that the problem has a solution if and only if λ 1 λ 0 . In particular, we discuss how to compute λ 0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.
The inverse spectrum problem for positive generalized stochastic matrices
Computers & Mathematics with Applications, 2002
consider the inverse spectrum problem for nonnegative matrices. In particular, we derive sufficient conditions for the existence of nonnegative and positive generalized stochastic and generalized doubly stochastic matrices with complex and real prescribed spectrum.