An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors
An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors
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DOI：
10.3934/naco.2013.3.583
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发表时间：
2012-03
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影响因子：
--
通讯作者：
Lixing Han
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文献类型：
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作者：
Lixing Han
设n为正整数，m为正偶数。设${\mathcal A}$是一个$m^{th}$阶$n维实弱对称张量，${\mathcal B}$是一个相同大小的实弱对称正定张量。$\lambda \in \mathbb{R}$被称为${\mathcal B}_r$- ${\mathcal a}$的特征值$如果${\mathcal a} x^{m-1} = \lambda {\mathcal B} x^{m-1}$对于某些$x \in \mathbb{R}^n \反斜杠\{0\}$。本文引入了两个无约束优化问题，得到了${\mathcal B}_r$的最小值和最大值——${\mathcal A}$的特征值的一些变分刻画。我们的结果推广了Auchmuty关于实对称矩阵特征值的无约束变分原理。这种无约束优化方法可用于寻找偶阶弱对称张量的Z、H或d特征值。我们提供了一些数值结果来说明这种方法在寻找z特征值和确定偶数阶对称张量的正半正定性方面的有效性。
Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size. $\lambda \in \mathbb{R}$ is called a ${\mathcal B}_r$-eigenvalue of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ for some $x \in \mathbb{R}^n \backslash \{0\}$. In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum ${\mathcal B}_r$--eigenvalues of ${\mathcal A}$. Our results extend Auchmuty's unconstrained variational principles for eigenvalues of real symmetric matrices. This unconstrained optimization approach can be used to find a Z-, H-, or D-eigenvalue of an even order weakly symmetric tensor. We provide some numerical results to illustrate the effectiveness of this approach for finding a Z-eigenvalue and for determining the positive semidefiniteness of an even order symmetric tensor.