Nonnegative Biquadratic Tensors
Chunfeng Cui, Liqun Qi
TL;DR
The paper extends $M$-eigenvalue theory to nonnegative biquadratic tensors by introducing $M^+$- and $M^{++}$-eigenvalues for NBQ$(m,n)$. It proves existence of at least one $M^+$-eigenvalue, establishes that the largest $M^+$-eigenvalue equals the largest $M$-eigenvalue and the $M$-spectral radius $ ho_M(oldsymbol{igA})$, and shows irreducible NBQ tensors have all $M^+$-eigenvalues as $M^{++}$-eigenvalues. It derives max-min characterizations for the largest $M^+$-eigenvalue and min-max for the smallest, and proposes a Collatz-type algorithm to compute the largest $M^+$-eigenvalue, complemented by numerical results. The work provides a practical spectral-analytic framework for nonnegative biquadratic tensors with potential applications in graph-related and statistical tensor problems.
Abstract
An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M$^+$-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors is positive, then that M$^+$-eigenvalue is called an M$^{++}$-eigenvalue. A nonnegative biquadratic tensor has at least one M$^+$ eigenvalue, and the largest M$^+$-eigenvalue is both the largest M-eigenvalue and the M-spectral radius. For irreducible nonnegative biquadratic tensors, all the M$^+$-eigenvalues are M$^{++}$-eigenvalues. Although the M$^+$-eigenvalues of irreducible nonnegative biquadratic tensors are not unique in general, we establish a sufficient condition to ensure their uniqueness. For an irreducible nonnegative biquadratic tensor, the largest M$^+$-eigenvalue has a max-min characterization, while the smallest M$^+$-eigenvalue has a min-max characterization. A Collatz algorithm for computing the largest M$^+$-eigenvalues is proposed. Numerical results are reported.