基于T-积的T-若尔当典范形与T-Drazin逆
T-Jordan Canonical Form and T-Drazin Inverse based on the T-Product
摘要：
本文引入张量相似变换，给出了T-若尔当典范形及其性质。提出了T-极小多项式和T-特征多项式的概念。作为一种特殊情况，我们提出的性质时，两个张量通过张量T-产品交换。凯莱-汉密尔顿定理也适用于张量的情况。然后重点讨论了张量分解理论。得到了张量的T-极分解、T-LU分解、T-QR分解和T-Schur分解。当F-平方张量通过T-积不可逆时，我们给出了T-群逆和T-Drazin逆，它们可以看作是矩阵情形的推广。利用T-若尔当典范形给出了T-群和T-Drazin逆的表达式.给出了T-Drazin逆的多项式形式。作为应用，我们提出了T-线性系，并求出了它的解.最后，我们给出了T-核-幂零分解，并证明了T-指标和T-Drazin逆可以由一个极限过程给出。
英文摘要：
In this paper, we introduce the tensor similar transforation and propose the T-Jordan canonical form and its properties. The concept of T-minimal polynomial and T-characteristic polynomial are raised. As a special case, we present properties when two tensors commutes via the tensor T-product. The Cayley-Hamilton theorem also holds for tensor cases. Then we focus on the tensor decomposition theory. T-polar, T-LU, T-QR and T-Schur decomposition of tensors are obtained. When a F-square tensor is not invertible via the T-product, we give the T-group inverse and T-Drazin inverse which can be viewed as the extension of matrix cases. The expression of T-group and T-Drazin inverse are given by the T-Jordan canonical form. The polynomial form of T-Drazin inverse is also proposed. As an application we raise the T-linear system and get its solution. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverse can be given by a limitation process.
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