Foundations of Economic Analysis

In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O’s and l’s. The l’s correspond to the arcs of the digraph. Certain matrix operations will be seen to correspond to digraph concepts.

Graphs and Matrices

Minimizing Kirchhoff index among graphs with a given vertex bipartiteness

Generalized tensor function via the tensor singular value decomposition based on the T-product

In this paper, we investigate the tensor similarity and propose the T-Jordan canonical form and its properties. The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed. As a special case, we present properties when two tensors commute based on the tensor T-product. We prove that the Cayley–Hamilton theorem also holds for tensor cases. Then, we focus on the tensor decompositions: T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained. When an F-square tensor is not invertible with the T-product, we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases. The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form. The polynomial form of the T-Drazin inverse is also proposed. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process.

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

In this paper, we define the Moore–Penrose inverse of tensors with the Einstein product, and the explicit formulas of the Moore–Penrose inverse of some block tensors are obtained. The general solutions of some multilinear systems are given and we also give the minimum-norm least-square solution of some multilinear systems using the Moore–Penrose inverse of tensors.

Moore–Penrose inverse of tensors via Einstein product

In this paper, we obtain formulas for resistance distances and Kirchhoff index of subdivision graphs. An application of resistance distances to the bipartiteness of graphs is given. We also give an interlacing inequality for eigenvalues of the resistance matrix and the Laplacian matrix.

Some results on resistance distances and resistance matrices

Brualdi-type eigenvalue inclusion sets of tensors

Some characterizations of M-tensors via digraphs

Let $\mathbb{K}$$^{m \times n}$ be the set of all the $m\times n$ matrices over the skew field. For the matrices $P,~Q \in $$\mathbb{K}$$^{n \times n}$, the explicit formulas for the Drazin inverse of $P + Q$ are given when $PQ^2 = 0$, $P^2QP = 0$, $(QP)^2 = 0$ and $P^2 QP = 0$, $P^3 Q = 0$, $Q^2 = 0$, respectively. Using these formulas, the representations for the Drazin inverse of the block matrices $ \left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right)\in\mathbb{K}^{n \times n}$ are showed with some conditions, where $A$ and $D$ are square.

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Lizhu Sun

Papers

Moore–Penrose inverse of tensors via Einstein product

Some results on resistance distances and resistance matrices

Brualdi-type eigenvalue inclusion sets of tensors

Some characterizations of M-tensors via digraphs

Formulas for the drazin inverse of matrices over skew fields