四阶张量Z-特征值的一个新的定位集及其应用

A new Z-eigenvalue localization set for fourth-order tensors and its application

偶数阶张量Z-特征值的定位在多元多项式的正定性判定中具有重要应用.本文研究4阶张量的Z-特征值定位问题,在一定条件下得到了4阶张量Z-特征值的一个新的包含集,改进了一些现有结果,并得到了4阶实对称张量正定性判定的一个易于验证的充分条件.最后通过数值例子说明其在4阶多项式的正定性判定中的应用.

The location of Z-eigenvalues of even order tensors plays an important role in determining the positive definiteness of multivariate polynomials.In this paper,the Z-eigenvalue localization of tensor of order 4 is studied.Under certain conditions,a new inclusion set of Z-eigenvalues of tensors of order 4 is obtained,which improves some existing results.A sufficient condition for the positive definiteness of real symmetric tensors of order 4 is given.Finally,a numerical example is given to illustrate its application in determining the positive definiteness of 4-order polynomials.