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Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

100%

Kurata H., Bapat R. B.

Special Matrices

2016

tom 4

nr 1

270-282

By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D+ in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B+ turns out to be the Laplacian of the tree, and the formula for D+ extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.

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

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.

Ograniczanie wyników

2 Special Matrices

1 Bapat R. B.

1 Johnson C. R.

1 Kurata H.

1 Toyonaga K.

1 2017

1 2016

Wyniki wyszukiwania

Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

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