SIAM Journal on Matrix Analysis and Applications· 2014Q1
$M$-Tensors and Some Applications
- 288citations
- Q1SCImago
- 2014year
Short summary
Researchers introduce $M$-tensors, an extension of $M$-matrices, demonstrating they must be $Z$-tensors with nonnegative maximal diagonal entries and that their minimal real eigenvalue is the smallest H$^+$-eigenvalue.
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Key points
- Introduced $M$-tensors as an extension of $M$-matrices.
- $M$-tensors are shown to be $Z$-tensors with nonnegative maximal diagonal entries.
- The minimal real part of an $M$-tensor's eigenvalues is its smallest H$^+$-eigenvalue.
- A criterion is established: a $Z$-tensor is an $M$-tensor iff all its H$^+$-eigenvalues are nonnegative.
- Strong $M$-tensors are defined, with all their H-eigenvalues being positive, applied to positive definiteness of multivariate forms.
AI-generated from the title and abstract; the full text is not read.
Abstract
We introduce $M$-tensors. This concept extends the concept of $M$-matrices. We denote $Z$-tensors as the tensors with nonpositive off-diagonal entries. We show that $M$-tensors must be $Z$-tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric $M$-tensor must be nonnegative. A symmetric $M$-tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an $M$-tensor is its smallest H$^+$-eigenvalue and also is its smallest H-eigenvalue. We show that a $Z$-tensor is an $M$-tensor if and only if all its H$^+$-eigenvalues are nonnegative. Some further spectral properties of $M$-tensors are given. We also introduce strong $M$-tensors, and some corresponding conclusions are given. In particular, we show that all $H$-eigenvalues of strong $M$-tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with $Z$-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form.
The authors' abstract, as published at the source. SIAM Journal on Matrix Analysis and Applications, 2014 · DOI ↗
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Field: Computational Mathematics
Computational MathematicsMathematics