|
Spectral properties of odd-bipartite Z-tensors and their absolute tensors
Haibin CHEN,Liqun QI
Front. Math. China. 2016, 11 (3): 539-556.
https://doi.org/10.1007/s11464-016-0520-4
Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Z-tensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.
References |
Related Articles |
Metrics
|
|
Generalized Vandermonde tensors
Changqing XU,Mingyue WANG,Xian LI
Front. Math. China. 2016, 11 (3): 593-603.
https://doi.org/10.1007/s11464-016-0528-9
We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor A=(Ai1i2...im) a type-1 generalized Vandermonde (GV) tensor, or GV1 tensor, if there exists a vector v=(v1,v2...vn)T such that Ai1i2...im=vi1i2+i3+...+im-m+1, and call A a type-2 (mth order ndimensional) GV tensor, or GV2 tensor, if there exists an (m-1)th order tensor B=(Bi1i2...im-1) such that Ai1i2...im=Bi1i2...im-1im-1. In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.
References |
Related Articles |
Metrics
|
|
lk,s-Singular values and spectral radius of partially symmetric rectangular tensors
Hongmei YAO,Bingsong LONG,Changjiang BU,Jiang ZHOU
Front. Math. China. 2016, 11 (3): 605-622.
https://doi.org/10.1007/s11464-015-0494-7
The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of lk,s-singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,s-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,ssingular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.
References |
Related Articles |
Metrics
|
|
Largest adjacency, signless Laplacian, and Laplacian H-eigenvalues of loose paths
Junjie YUE,Liping ZHANG,Mei LU
Front. Math. China. 2016, 11 (3): 623-645.
https://doi.org/10.1007/s11464-015-0452-4
We investigate k-uniform loose paths. We show that the largest Heigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥3, we show that the largest H-eigenvalue of its adjacency tensor is ((1+5)/2)2/k when l=3 and λ(A)=31/k when l=4, respectively. For the case of l≥5, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval (2, 3) when l≥5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.
References |
Related Articles |
Metrics
|
17 articles
|