Some Research On Generalized Jordan Canonical Form

The Jordan canonical form of matrix is an important concept in linear algebra.In recent years,with the rapid development of tensor analysis,scholars have proposed the tensor Jordan canonical form in the low-rank approximation of tensors.Because of the complexity and diffi-culty of this problem,most of the present research results are about the Jordan canonical form of low-order and low-dimensional tensors.In this paper,we mainly study Jordan canonical form of three-way tensor.The paper is divided into three chapters:In Chapter 1,we mainly introduces the development of tensor Jordan canonical form,and the basic conceptions and the research status.In Chapter 2,we study the Jordan canonical form of the tensor when the upper triangular elements of the tensor has zero elements.If the upper triangular elements of the last three slices of tensor gj has zeros,then gj may not be transformed into Jordan canonical form.In this case,we study the conditions that guarantee tensor gj can be turned into canonical form.When some upper triangular elements elements are zeros,we first study all combinations of these elements,and obtain 91 combinations.Then we divide 91 combinations into three sets T1,T2,T3 and prove that:for each combinations of T1,there always exists two slices in the last three slices whose second diagonal elements can be turned to zeros;for each combinations of T2,the second diagonal elements of any two slices of the last three slices cannot be turned into zeros.Because of the complexity of the combinations in T3,we have not yet analyzed the combinations in T3.Finally,for each combinations in T1,we continue to analyze the conditions that guarantee the third and fourth diagonal elements can be turned into zeros,i.e.the conditions that guarantee tensor gj can be turned into canonical form.In Chapter 3,we study the Jordan canonical form of tensor with multilinear rank(4,4,3).For the three-way tensor gj with multilinear rank(4,4,3),we first give its Jordan canonical form and prove that:if all the upper triangular elements in the last three slices are nonzero,we show that gj must be turned into Jordan canonical form;otherwise,gj may not be turned into Jordan canonical form.In this case,we give the conditions that guarantee gj can be turned into canonical form.