Diagonalizable higher degree forms and symmetric tensors
Diagonalizable higher degree forms and symmetric tensors
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DOI：
10.1016/j.laa.2020.12.018
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发表时间：
2020-03
影响因子：
1.1
通讯作者：
Hua-Lin Huang;Huajun Lu;Yu Ye;Chi Zhang
中科院分区：
文献类型：
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作者：
Hua-Lin Huang;Huajun Lu;Yu Ye;Chi Zhang
我们提供了简单的标准和算法表示齐次多项式作为独立的线性形式的幂和，或等价地，分解成线性无关向量的秩1对称张量的对称张量的总和。该准则依赖于两个方面的高次形式，即哈里森的代数理论和一些代数几何性质。所提出的算法是纯粹基于求解线性和二次方程。此外，作为我们的标准和算法的副产品，人们可以很容易地决定是否齐次多项式或对称张量是正交或酉可分解的。
We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent vectors. The criteria rely on two facets of higher degree forms, namely Harrison's algebraic theory and some algebro-geometric properties. The proposed algorithms are based purely on solving linear and quadratic equations. Moreover, as a byproduct of our criteria and algorithms one can easily decide whether or not a homogeneous polynomial or symmetric tensor is orthogonally or unitarily decomposable.