Further results on generalized inverses of tensors via the Einstein product

Abstract The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.

[1]  Michael Greenacre,et al.  Multiway data analysis , 1992 .

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

[3]  David Rubin,et al.  Introduction to Continuum Mechanics , 2009 .

[4]  Nikos D. Sidiropoulos,et al.  Parallel factor analysis in sensor array processing , 2000, IEEE Trans. Signal Process..

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

[6]  Lars Grasedyck,et al.  Existence and Computation of Low Kronecker-Rank Approximations for Large Linear Systems of Tensor Product Structure , 2004, Computing.

[7]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[8]  Linzhang Lu,et al.  A projection method and Kronecker product preconditioner for solving Sylvester tensor equations , 2012 .

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

[10]  K. S. Banerjee.  Generalized Inverse of Matrices and Its Applications , 1973 .

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

[12]  Alan J. Mayne,et al.  Generalized Inverse of Matrices and its Applications , 1972 .

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

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

[15]  Li Qiu,et al.  Unitary dilation approach to contractive matrix completion , 2004 .

[16]  Ben-Wen Li,et al.  Chebyshev Collocation Spectral Methods for Coupled Radiation and Conduction in a Concentric Spherical Participating Medium , 2009 .

[17]  Ben-Wen Li,et al.  Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method , 2010, J. Comput. Phys..

[18]  Nikos D. Sidiropoulos,et al.  Blind PARAFAC receivers for DS-CDMA systems , 2000, IEEE Trans. Signal Process..

[19]  Charles Van Loan,et al.  Block Tensor Unfoldings , 2011, SIAM J. Matrix Anal. Appl..