Perron–Frobenius theorem for nonnegative multilinear forms and extensions

By Shmuel Friedland, Stephane Gaubert, Lixing Han. 2009

393 citations recorded by the source corpus.

We prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/3d834ea31f2e885be6471d60638abd414cb266e0

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. J. Shao
5. Guanglu Zhou
6. L. Caccetta
7. H. Shan
8. Zhen Chen

In Computational Theory and Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Guanglu Zhou
4. Liqun Qi
5. J. Shao
6. L. Caccetta
7. H. Shan
8. Zhen Chen

In Numerical Analysis

1. Vladimir Nikiforov
2. L. Caccetta
3. J. Shao
4. Chaoqian Li
5. Guanglu Zhou
6. Liqun Qi
7. Yaotang Li
8. Xiaojun Chen

Citations in the corpus

• Symmetric nonnegative tensors and copositive tensors (2012, background): “By [5,9], there is a partition (I1, · · · , Is) of {1, · · · , n}, such that A(Ir) is weakly irreducible for r = 1, · · · , s, and ai1···ik = 0 for all i1 ∈ Ir, i2, · · · , ik ∈ Ir, r = 1, · · · , s.”
• Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs (2014, background): “It was proved in [6,7] that a k-hypergraph G is connected if and only if its adjacency tensor A(G) is weakly irreducible.”
• H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors (2013, methodology): “There is a rich theory on eigenvalues of a nonnegative tensor [4, 5, 8, 17, 18, 27, 28].”
• On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs (2014, background): “[6] proposed a weak version of irreducible nonnegative tensors T .”
• On the limiting probability distribution of a transition probability tensor (2014, neutral): “6 [15,17,18] If A is an irreducible non-negative tensor of order m and dimension n, then there exist λ0 > 0 and x̄ > 0 with x̄ ∈ Rn such that Ax̄m−1 = λ0x̄[m−1].”
• M-Tensors and Some Applications (2014, neutral): “If A is not weakly reducible, then it is called weakly irreducible [12, 16].”
• Z-EIGENVALUE INCLUSION THEOREMS FOR TENSORS (2016, background): “There is abundant literature on the development of H-eigenvalues for nonnegative tensors [2, 4, 8, 9, 12, 14, 20, 22, 23].”