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It is known that an n-by-n Hermitian matrix, n ≥ 2 , whose graph is a tree necessarily has at least two eigenvalues (the largest and smallest, in particular) with multiplicity 1. Recently, much of the multiplicity theory, for eigenvalues of Hermitian matrices whose graph is a tree, has been generalized to geometric multiplicities of eigenvalues of matrices over a general field (whose graph is a tree). However, the two 1's fact does not generalize. Here, we give circumstances under which there are two 1's and give several examples (without two 1's) that limit our positive results.

Given a tree T , a Hermitian matrix A whose graph is T and an eigenvalue λ of A , a number of new structural results about T relative to the multiplicity of λ in A are developed. These include a complete classification of the possible changes in status of one vertex upon removal of another. These are used, in part, to give a unique fundamental decomposition of the tree that can be used to answer further structural questions. In the process, the notions of singly and multiply Parter vertices are introduced and used. Possible changes in the structure of the fundamental decomposition, resulting from changes in one diagonal entry, or deletion of a row and a column, are also discussed.

For an Hermitian matrix A whose graph is a tree T , we study the number of eigenvalues of A whose multiplicity decreases when a particular vertex is deleted from T . Explicit results are given when that number of eigenvalues is less than 4 and an inductive result thereafter. The work is based, in part, on classical results about multiplicities, but also on some new facts, including a useful identity. This allows us to give strong bounds based on simple facts about the location of the vertex in the tree. Some facts about matrices whose graphs are not trees are included, and the classical diameter bound about the number of distinct eigenvalues for a tree follows.

A class of m -by- n real (complex) matrices is semi-algebraic (SA) if membership in it may be checked via a finite list of polynomial inequalities in the entries (the real and imaginary parts of the entries). Real algebraic geometry is used to show that (1) very many familiar classes of matrices (some perhaps surprising) are SA and (2) that two important derived problems (matrix completions and Hadamard Duals) lead to an SA solution when they come from an SA class. This means that at least there exists a finite solution to these problems though it may be difficult to find. The methodology likely extends to other problems of interest.

In the last decade, the potential of mobile devices for augmenting outdoor experience opened up new solutions, whose value is twofold. On one hand, users can experience new forms of interaction with space. On the other, stakeholders can have access to the so-called User Generated Data, that is different types of information related to public spaces that could be used to improve their conception of space. In line with this, several digital tools have been developed and tested within the framework of CyberParks COST Action TU-1306 with the intention of exploring how information and communication technologies (ICTs) can contribute to the improvement of Public Open Spaces (POS). In this way, this chapter aims to study the relationship between ICTs and POS, focused on the opportunities offered by three different digital tools: the WAY-CyberParks, the EthnoAlly, and the CyberCardeto. The main advantages of using these digital tools are: (1) the real-time data gathering, (2) maintaining an updated database, (3) collecting traces of different activities and users’ groups “at the same time and space”, and (4) recording their opinion and preferences, via text, video, sound or pictures. Furthermore, the chapter attempts to analyse distinct types of results produced by their use, based on different study cases where the digital tool has been tested. The data obtained in these places serve to demonstrate the features and type of data gathered. With these case studies, the chapter attempts to highlight the main potential of each platform as related to different stakeholders and users. This work has been supported by the Cost Action TU1306 CyberParks and the Spanish Ministry of Economy and Competitiveness under the ESPHIA project (TIN2014-56042-JIN). E. Osaba would like to thank the Basque Government for its funding support through the EMAITEK program. The authors would also like to thank the staff of UbiSive s.r.l. for the support in developing the CyberCardeto smartphone application.

We investigate the relation between perturbing the i-th diagonal entry of A ∈ ℳ n (𝔽) and extracting the principal submatrix A(i) from A with respect to the possible changes in multiplicity of a given eigenvalue. A complete description is given and used to both generalize and improve prior work about Hermitian matrices whose graph is a given tree.

When can an ( n - k ) × ( n - k ) normal matrix B be imbedded in an n × n normal matrix A? This question was studied for the first time 50 years ago by Ky Fan and Gordon Pall, who gave the complete answer in the case k = 1 . Since then, a few authors obtained additional results. In this note, we show how an approach inspired by the Hermitian case can throw some light on the problem.

Through a succession of results, it is known that if the graph of an Hermitian matrix A is a tree and if for some index j, λ∈σ(A)∩σ(A(j)), then there is an index i such that the multiplicity of λ in σ(A(i)) is one more than that in A. We exhibit a converse to this result by showing that it is generally true only for trees. In particular, it is shown that the minimum rank of a positive semidefinite matrix with a given graph G is ⩽n-2 when G is not a tree. This raises the question of how the minimum rank of a positive semidefinite matrix depends upon the graph in general.

There is a symmetric nonnegative matrix A, subordinate to a given bipartite graph G on n vertices, with eigenvalues λ1≥λ2≥≥λ n if and only if, λ1 + λ n ≥0, λ2 + λ n-1≥0,...,λ m + λ n - m + 1≥0, λ m + 1≥0,...,λ n - m ≥0, in which m is the matching numberof G. Other observations are also made about the symmetric nonnegative inverse eigenvalue problem with respect to a graph

For Hermitian matrices, whose graph is a given tree, the relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail. In the process, certain aspects of special vertices, whose removal results in an increase in multiplicity are investigated.

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.

Sem PDF conforme despacho. For a given graph, there is a natural question of the possible lists of multiplicities for the eigenvalues among the spectra of Hermitian matrices with that graph (no constraint is placed upon the diagonal entries of the matrices by the graph). Here, we survey some of what is known about this question and include some new information about it. There is a natural focus upon the case in which the graph is a tree. In this event, there is remarkable structure to the possible lists. Both the general theory and a summary of specific results is given. At the end, this allows to give, in compact tabular form, all lists for trees on fewer than 11 vertices (a potentially valuable tool for further work). There is a brief discussion of non-trees. publishersversion published

An important theorem about the existence of principal submatrices of a Hermitian matrixwhose graph is a tree, in which the multiplicity of an eigenvalue increases, was largely developed in separate papers by Parter and Wiener. Here, the prior work is fully stated, then generalized with a self-contained proof. The more complete result is then used to better understand the eigenvalue possibilities of reducible principal submatrices of Hermitian tridiagonal matrices. Sets of vertices, for which the multiplicity increases, are also studied. publishersversion published

We characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star (a tree in which at most one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible ordered multiplicities is equivalent to the inverse eigenvalue problem for a given tree. Moreover, a key spectral feature of the inverse eigenvalue problem in the case of generalized stars is shown to characterize them among trees.

The main result of this paper is the following Let A and B be n×n hermitian matrices with eigenvalues respectively, ordered so that and let M1 be any k×k principal submatrix of . Necessary and sufficient conditions for equality are given.

Given real numbers λ 1 ,…,λ n ,μ 1 ,…,μ n −1 with λ 1 ⩾μ 2 ⩾λ 2 ⩾ … ⩾μ n −1⩾ λ n , an integer i with 1⩽ i ⩽ n , and a tree Γ with n vertices, we prove the existence of a Hermitian acyclic matrix A =[ x rs ] with eigenvalues λ 1 ,…,λ n and x rs =0 whenever r ≠ s and { r , s } is not an edge of Γ and such that the principal submatrix of A obtained from A by deleting the i th row and column has eigenvalues μ 1 ,…,μ n −1 .

Fiedler proved in [Linear Algebra Appl. 2 (1969) 191–197] that the set of real n-by-n symmetric matrices A such that rank(A + D) ⩾ n − 1 for every real diagonal matrix D is the set of matrices PTPT where P is a permutation matrix and T an irreducible tridiagonal matrix. We show that this result remains valid for arbitrary fields with some exceptions for 5-by-5 matrices over Z 3 .

There is remarkable and distinctive structure among Hermitian matrices, whose graph is a given tree T and that have an eigenvalue of multiplicity that is a maximum for T. Among such structure, we give several new results: (1) no vertex of T may be "neutral"; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest. http://www.sciencedirect.com/science/article/B6V0R-4SMF2K9-2/1/e80ee40f33898e52f517c5d58dbfa5bc

We consider the general problem of determining which lists of multiplicities for the eigenvalues occur among Hermitian matrices the graph of whose off-diagonal entries is a given tree. Several restrictions are cited and a construction strategy is given. Together, these are sufficient to characterize all lists for each tree in two infinite classes: the double paths and generalized stars, and to tabulate all lists for trees on fewer than nine vertices. Such tables should be useful for formulating and dispelling general conjectures.