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Sufficient conditions to be exceptional

100%
Johnson C. R., Reams R. B.
Special Matrices
|
2016
|
tom 4
|
nr 1
67-72
EN
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).
2
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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

100%
Toyonaga K., Johnson C. R.
Special Matrices
|
2017
|
tom 5
|
nr 1
51-60
EN
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 Parter vertices. Then, we investigate the change in multiplicity of an eigenvalue based upon a change in an edge value. We show how the multiplicity of the eigenvalue changes depending upon the status of the edge and the edge value. This work explains why, in some cases, edge values have no effect on multiplicities. We also characterize, more precisely, how multiplicity changes with the removal of two adjacent vertices.
3
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Ranks of permutative matrices

81%
Hu X., Johnson C. R., Davis C. E., Zhang Y.
Special Matrices
|
2016
|
tom 4
|
nr 1
233-246
EN
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.
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