The Drazin inverse of an even-order tensor and its application to singular tensor equations

Abstract The notion of the Moore–Penrose inverses of matrices was recently extended from matrix space to even-order tensor space with Einstein product in the literature. In this paper, we further study the properties of even-order tensors with Einstein product. We define the index and characterize the invertibility of an even-order square tensor. We also extend the notion of the Drazin inverse of a square matrix to an even-order square tensor. An expression for the Drazin inverse through the core-nilpotent decomposition for a tensor of even-order is obtained. As an application, the Drazin inverse solution of the singular linear tensor equation A ∗ X = B will also be included.

[1]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[2]  Lars Elden,et al.  Matrix methods in data mining and pattern recognition , 2007, Fundamentals of algorithms.

[3]  Linda B. McGown,et al.  Resolution of multicomponent fluorescent mixtures by analysis of the excitation–emission–frequency array , 1990 .

[4]  Na Li,et al.  Solving Multilinear Systems via Tensor Inversion , 2013, SIAM J. Matrix Anal. Appl..

[5]  Yimin Wei,et al.  Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product , 2017 .

[6]  J. Shao.  A general product of tensors with applications , 2012, 1212.1535.

[7]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[8]  Yiju Wang,et al.  Some advances in tensor analysis and polynomial optimization , 2014 .

[9]  Yimin Wei.  Index splitting for the Drazin inverse and the singular linear system , 1998, Appl. Math. Comput..

[10]  Y. Ye,et al.  Linear operators and positive semidefiniteness of symmetric tensor spaces , 2015 .

[11]  A. Einstein.  The Foundation of the General Theory of Relativity , 1916 .

[12]  Hebing Wu,et al.  Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index , 2000 .

[13]  Changjiang Bu,et al.  Moore–Penrose inverse of tensors via Einstein product , 2016 .

[14]  Hanspeter Pfister,et al.  Face transfer with multilinear models , 2005, SIGGRAPH 2005.

[15]  Changjiang Bu,et al.  The inverse, rank and product of tensors , 2014 .

[16]  Liqun Qi,et al.  Algebraic connectivity of an even uniform hypergraph , 2012, J. Comb. Optim..

[17]  Joshua N. Cooper,et al.  Spectra of Uniform Hypergraphs , 2011, 1106.4856.

[18]  Debasisha Mishra,et al.  Further results on generalized inverses of tensors via the Einstein product , 2016, 1604.02675.

[19]  J. Shao,et al.  On some properties of the determinants of tensors , 2013 .

[20]  Predrag S. Stanimirovic,et al.  Gradient methods for computing the Drazin-inverse solution , 2013, J. Comput. Appl. Math..

[21]  Julio Benítez,et al.  The generalized inverses of tensors and an application to linear models , 2017, Comput. Math. Appl..

[22]  Qingzhi Yang,et al.  Further Results for Perron-Frobenius Theorem for Nonnegative Tensors , 2010, SIAM J. Matrix Anal. Appl..

[23]  Yimin Wei,et al.  Fast Hankel tensor–vector product and its application to exponential data fitting , 2015, Numer. Linear Algebra Appl..

[24]  Pierre Comon,et al.  Tensor Decompositions, State of the Art and Applications , 2002 .

[25]  Yimin Wei,et al.  Theory and Computation of Tensors: Multi-Dimensional Arrays , 2016 .

[26]  Changjiang Bu,et al.  E-cospectral hypergraphs and some hypergraphs determined by their spectra , 2014, 1406.1085.

[27]  Rasmus Bro,et al.  Multi-way Analysis with Applications in the Chemical Sciences , 2004 .