Spectra inhabiting the left half-plane that are universally realizable

Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

Centrosymmetric universal realizability

A list $\Lambda =\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $\Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $\Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $\Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.

SIMETRISASI BENTUK KANONIK JORDAN

If the characteristic polynomial of a linear operator is completely factored in scalar field of then Jordan canonical form of can be converted to its rational canonical form of , and vice versa. If the characteristic polynomial of linear operator is not completely factored in the scalar field of ,then the rational canonical form of can still be obtained but not its Jordan canonical form matrix . In this case, the rational canonical form of can be converted to its Jordan canonical form by extending the scalar field of to Splitting Field of minimal polynomial of , thus forming the Jordan canonical form of over Splitting Field of . Conversely, converting the Jordan canonical form of over Splitting Field of to its rational canonical form uses symmetrization on the Jordan decomposition basis of so as to form a cyclic decomposition basis of which is then used to form the rational canonical matrix of

A NEW ALGORITHM FOR DECOMPOSING MODULAR TENSOR PRODUCTS

Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$ , the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$ . This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$ . Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$ . Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.

Permutative universal realizability

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.

Spectra of products of digraphs

A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues.

The role of certain Brauer and Rado results in the nonnegative inverse spectral problems

It is said that a list $\Lambda =\{\lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$. It is said that $\Lambda $ is universally realizable if it is realizable for each possible Jordan canonical form allowed by $\Lambda$. This work does not contain new results. As its title says, its goal is to show and emphasize the relevance and importance of certain results, by Brauer and Rado, in the study of nonnegative inverse spectral problems. It is shown that virtually all known results, which give sufficient conditions for $\Lambda$ to be realizable or universally realizable, can be obtained from results by Brauer and Rado. Moreover, from these results, a realizing matrix may always be constructed.