Further Results for Perron-Frobenius Theorem for Nonnegative Tensors

By Yuning Yang, Qingzhi Yang. SIAM Journal on Matrix Analysis and Applications, 2010

461 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/edfdc443281cc95055f3be014e1986fb6e3bed48

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Liqun Qi
4. Yisheng Song
5. Guanglu Zhou
6. L. Caccetta
7. Aiquan Jiao
8. Yiju Wang

In Computational Theory and Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Liqun Qi
4. Yisheng Song
5. Guanglu Zhou
6. L. Caccetta
7. Yiju Wang
8. Zhen Chen

Citations in the corpus

• Nonsingular $H$-tensor and its criteria (2016, background): “However, its conditions are hard to verify and here we introduce the following popular condition in tensor analysis [28, 34, 36].”
• Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process (2013, background): “ On the other hand, A(ǫ) is positive, which is irreducible in the sense of [2]. According to the Perron-Frobenius theorem [2], λ(ǫ)0 is the largest eigenvalue of A(ǫ). And the proof of Theorem 2.3 of [15] shows that λ0 = limǫ→0 λ(ǫ)0 is the largest eigenvalue of A since A = limǫ→0 A(ǫ). ✷ Suppose C ”
• An S-type eigenvalue localization set for tensors (2015, background): “This result can be easily generalized to general tensors [13, 26].”
• An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors (2016, background): “A real tensor A = (ai1···im ) is called symmetric [1,2] if ai1···im = aπ(i1···im), ∀π ∈ m, where m is the permutation group of m indices.”
• A new eigenvalue inclusion set for tensors and its applications (2015, background): “However, many important results on the eigenvalue problem of matrices have been successfully extended to higher order tensors; see [1,2,13, 15,18,19,21,26,27].”
• M-Tensors and Some Applications (2014, unknown): “Then, we have this lemma given in [26].”
• Infinite and finite dimensional Hilbert tensors (2014, result): “[17] Q. Yang and Y. Yang, Further Results for Perron-Frobenius Theorem for Nonnegative Ten- sors II, SIAM.”
• Symmetric nonnegative tensors and copositive tensors (2012, background): “[19] If A is a nonnegative tensor of order k and dimension n, then ρ(A) is an H+-eigenvalue of A.”
• Z-EIGENVALUE INCLUSION THEOREMS FOR TENSORS (2016, methodology): “In particular, H-eigenvalue inclusion theorems were established for general tensors [11, 14] and the bounds of the largest H-eigenvalue were given for nonnegative tensors [11, 14, 19, 20].”
• M-tensors and nonsingular M-tensors (2013, background): “If B is a nonnegative tensor of order m and dimension n, then ρ(B) is an eigenvalue of B with a nonnegative eigenvector x ∈R+ [16,17].”
• Z-tensors and complementarity problems (2015, background): “Properties of nonnegative tensors, particularly in relation to the Perron-Frobenius theorem, have been explored in several recent papers, see [3, 24, 23, 26].”
• Some upper bounds on the eigenvalues of uniform hypergraphs (2015, background): “(See [7,11].”