An even order symmetric B tensor is positive definite

By Liqun Qi, Yisheng Song. 2014

111 citations recorded by the source corpus.

It is easily checkable if a given tensor is a B tensor, or a B0 tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B0 tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B0 tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case.

Subfields: Computational Mathematics, Radiology, Nuclear Medicine and Imaging, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/8dee64c187af0dcad1cfb642e17ad04ac3bba958

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In Computational Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Yisheng Song
4. Haibin Chen
5. Zhongming Chen
6. Liqun Qi
7. J. Shao
8. Guanglu Zhou

In Radiology, Nuclear Medicine and Imaging

1. Chaoqian Li
2. Yaotang Li
3. Haibin Chen
4. Yisheng Song
5. Liqun Qi
6. Zhongming Chen
7. Aiquan Jiao

In Spectroscopy

1. Chaoqian Li
2. Yaotang Li
3. Liqun Qi

Citations in the corpus

• An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors (2016, background): “Before that we give the definition of partially all one tensors, which proposed by Qi and Song [28].”
• Double B-tensors and quasi-double B-tensors (2014, background): “[27] Let B = (ai1···im) be a real tensor of order m dimension n.”
• Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process (2013, background): “It was proved in [24] that an even order symmetric B tensor is positive definite and an even order symmetric B0 tensor is positive semi-definite.”
• Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors (2014, background): “In recent years, a lot of research papers on structured tensors are presented [2, 3, 4, 8, 13, 14, 15, 17, 18, 21, 22], which studied M-tensors, circulant tensors, completely positive tensors, Hankel tensors, Hilbert tensors, P tensors and B tensors.”
• Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors (2014, background): “Furthermore, Song and Qi [18] studied some nice properties of such a class of tensors.”