Nonnegative Biquadratic Tensors

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.