Currently displaying 1 – 20 of 20

Showing per page

Order by Relevance | Title | Year of publication

The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value

Kenji ToyonagaCharles R. Johnson — 2017

Special Matrices

We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue, when the edge is removed (i.e. the corresponding entry of A is replaced by 0).We show a necessary and suficient condition for each possible classification of an edge. A special relationship is observed among 2-Parter edges, Parter edges and singly...

Sufficient conditions to be exceptional

Charles R. JohnsonRobert B. Reams — 2016

Special Matrices

A copositive matrix A is said to be exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. We show that with certain assumptions on A−1, especially on the diagonal entries, we can guarantee that a copositive matrix A is exceptional. We also show that the only 5-by-5 exceptional matrix with a hollow nonnegative inverse is the Horn matrix (up to positive diagonal congruence and permutation similarity).

Applications of saddle-point determinants

Jan HaukeCharles R. JohnsonTadeusz Ostrowski — 2015

Discussiones Mathematicae - General Algebra and Applications

For a given square matrix A M n ( ) and the vector e ( ) n of ones denote by (A,e) the matrix ⎡ A e ⎤ ⎣ e T 0 ⎦ This is often called the saddle point matrix and it plays a significant role in several branches of mathematics. Here we show some applications of it in: game theory and analysis. An application of specific saddle point matrices that are hollow, symmetric, and nonnegative is likewise shown in geometry as a generalization of Heron’s formula to give the volume of a general simplex, as well as a conditions...

Ranks of permutative matrices

Xiaonan HuCharles R. JohnsonCaroline E. DavisYimeng Zhang — 2016

Special Matrices

A new type of matrix, termed permutative, is defined and motivated herein. The focus is upon identifying circumstances under which square permutative matrices are rank deficient. Two distinct ways, along with variants upon them are given. These are a special kind of grouping of rows and a type of partition in which the blocks are again permutative. Other, results are given, along with some questions and conjectures.

New results about semi-positive matrices

Jonathan DorseyTom GannonCharles R. JohnsonMorrison Turnansky — 2016

Czechoslovak Mathematical Journal

Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least 2 elements is the spectrum of a square semipositive matrix,...

Exponential polynomial inequalities and monomial sum inequalities in p -Newton sequences

Charles R. JohnsonCarlos MarijuánMiriam PisoneroMichael Yeh — 2016

Czechoslovak Mathematical Journal

We consider inequalities between sums of monomials that hold for all p-Newton sequences. This continues recent work in which inequalities between sums of two, two-term monomials were combinatorially characterized (via the indices involved). Our focus is on the case of sums of three, two-term monomials, but this is very much more complicated. We develop and use a theory of exponential polynomial inequalities to give a sufficient condition for general monomial sum inequalities, and use the sufficient...

Page 1 Next

Download Results (CSV)