p -Norm B -Tensors and p -Norm $$B_0$$ B 0 -Tensors
- PDF / 315,273 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 116 Downloads / 226 Views
p-Norm B-Tensors and p-Norm B0 -Tensors Qilong Liu1 · Qingshui Liao1 Received: 26 June 2019 / Revised: 27 October 2019 / Accepted: 26 November 2019 © Iranian Mathematical Society 2019
Abstract We propose two new classes of tensors: p-norm B-tensors and p-norm B0 -tensors, and discuss their relationships with P(P0 )-tensors and M B (M B0 )-tensors. We prove that a real symmetric p-norm B(B0 )-tensor can always be decomposed into the sum of a p-norm strictly diagonally dominant ( p-norm diagonally dominant) Z -tensor and several positive multiples of partially all one tensors. Specially, when the order of the tensor is even, we obtain that the corresponding real symmetric p-norm B(B0 )-tensor is positive (semi-)definite. This gives a checkable sufficient condition for the positive (semi-)definite tensors. Keywords p-Norm B-Tensors · p-Norm B0 -Tensors · Positive definite tensors · Sufficient condition Mathematics Subject Classification 47H15 · 15A18 · 65F15
1 Introduction Identifying the positive (semi-)definiteness of real symmetric tensors arises in various applications, such as automatical control [1,8,20,47], magnetic resonance imaging [5,10,41,42], spectral hypergraph theory [13–15,30,38], and polynomial problems [43,45]. The positive (semi-)definiteness of real symmetric tensors receives much attention. For instance, Hulsen presents a sufficient condition for the positive definiteness of a configuration tensor in the Giesekus model [16]. Based on the Sturm theorem, a numerical method is proposed for identifying the positive definiteness of a
Communicated by Abbas Salemi.
B
Qilong Liu [email protected] Qingshui Liao [email protected]
1
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
tensor with low dimensions [1]. However, when the dimension of the tensor is great than 3 and the order of the tensor is not less than 4, it is difficult to determine whether a given real symmetric tensor is positive definite or not since the problem is NP-hard [9]. For solving this problem, Qi [36] proposed the definition of eigenvalues and eigenvectors for tensors, and pointed out that an even-order real symmetric tensor is positive (semi-)definite if and only if all of its H-eigenvalue are positive (nonnegative). An eigenvalue method for identifying the positive definiteness of a multivariate form is given in [35]. Besides, there is research work with which one can exactly compute the largest H-eigenvalue for nonnegative irreducible tensors [4,33,34,49], nonnegative tensors [11,48,50,57,58], essentially positive tensors [53], weakly positive tensors [54] and essentially nonnegative tensors [12]. However, it is difficult to compute all Heigenvalues exactly when the order and the dimension of a tensor is large. To overcome this problem, Qi [36] extended the well-known Ger˘s gorin’s eigenvalue localization set of matrices to real symmetric tensors. For an even-order real symmetric tensor, if the eigenvalue localization
Data Loading...
