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On determinants and eigenvalue theory of tensors. (English) Zbl 1259.15038
The authors investigate properties of the determinants of tensors, and their applications in the eigenvalue theory of tensors. They show that the determinant inherits many properties of the determinant of a matrix. These properties include solvability of polynomial systems, a product formula for the determinant of a block tensor, a product formula of the eigenvalues and Geršgorin’s inequality. As a simple application, they show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution in the complex space. They also investigate the characteristic polynomial of a tensor through the determinant and the higher order traces and prove that the \(k\)th order trace of a tensor is equal to the sum of the \(k\)th powers of the eigenvalues of this tensor, and the coefficients of its characteristic polynomial are recursively generated by the higher order traces. An explicit formula for the second order trace of a tensor is also given.

MSC:
15A72 Vector and tensor algebra, theory of invariants
15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
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