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Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Published on: July 28, 2013

Permutation transformations of tensors with an application.

Yao-Tang Li¹, Zheng-Bo Li¹, Qi-Long Liu¹

¹School of Mathematics and Statistics, Yunnan University, Kunming, 650091 People's Republic of China.

|December 21, 2016

Summary

This summary is machine-generated.

This study introduces tensor permutation transformations and examines their invariance properties for various tensor structures. It presents a canonical form theorem, simplifying high-dimensional tensor problems by reducing them to lower-dimensional ones.

Keywords:

Canonical form Permutation transformation Structure tensor Weakly irreducible tensor

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Area of Science:

Multilinear Algebra
Numerical Analysis
Tensor Theory

Background:

Tensors are fundamental in various scientific fields, but high-dimensional tensor problems are computationally challenging.
Understanding tensor properties and transformations is crucial for developing efficient analytical methods.

Purpose of the Study:

To introduce the concept of permutation transformations for tensors.
To investigate the invariance properties of permutation transformations on key tensor structures.
To present a canonical form theorem for tensors as an application.

Main Methods:

Introduction and definition of tensor permutation transformations.
Analysis of invariance properties under these transformations for specific tensor classes (symmetric, positive definite, Z-tensors, M-tensors, Hankel tensors, P-tensors, B-tensors, H-tensors).
Development and presentation of the canonical form theorem for tensors.

Main Results:

The basic properties of tensor permutation transformations are established.
Invariance under permutation transformations is demonstrated for several important tensor types.
A canonical form theorem is derived, showing that high-dimensional tensor problems can be simplified.

Conclusions:

Permutation transformations offer a novel approach to tensor analysis.
The canonical form theorem provides a powerful tool for simplifying complex tensor problems.
This work facilitates easier handling and analysis of higher-dimensional tensor computations.