\(M\)-tensors and some applications (Q2923351)

Tensors appear increasingly in numerous diverse applications of mathematics. An \(m\)-order \(n\)-dimensional square tensor \(A\), where \(m, n\) are positive integers, is taken to be an array with \(nm\) real entries which becomes an \(n\times n\) matrix for \(m=2\). If all the off-diagonal entries of such a tensor \(A\) are nonpositive, then \(A\) is called a \(Z\)-tensor. Moreover, a tensor \(A\) is called an \(M\)-tensor if a nonnegative tensor \(B\) and a positive real number \(\eta\geq\rho(B)\), where \(\rho\) denotes the spectral radius, exist such that \(A=\eta I-B\). It follows that \(M\)-tensors are \(Z\)-tensors. The authors go on to make a detailed study of the spectral properties of \(M\)-tensors, leading to an easy method for determining whether a \(Z\)-tensor is an \(M\)-tensor, and a sufficient condition for a tensor to be an \(M\)-tensor. In the final section, they give some applications of \(M\)-tensors based on the earlier spectral results, including testing the positive definiteness of a certain multivariate form associated with a symmetric \(Z\)-tensor.