Belitskii’s Canonical Forms For Upper Triangular Nilpotent Matrices

Linear matrix problem introduced by V. V. Sergeichuk seems to be an elegant for-mulation of matrix problems. Roughly speaking, the so-called matrix problem is to study the similarity problem of a set of matrices under some admissible transforma-tions. One of its central issues is to find standard forms under similarity. Beliskii’s al-gorithm is an effective tool to calculate the canonical forms of any matrix under certain admissible transformations. Sergeichuk has built representation equivalence between the representation category of finite dimensional algebras and the matrix category of the corresponding linear matrix problem successfully. As a result, the classification of representations of finite dimensional algebras can be reduced to find canonical forms of indecomposable matrices of the corresponding linear matrix problem.The linear matrix problem in the thesis is about the canonical forms for n x n upper triangular matrices under upper triangular nilpotent similarity. It is known that the linear matrix problem for n≤5 has finitely similarity classes of indecomposable matrices; the linear matrix problem for n≥ 6 has infinitely many similarity classes of indecomposable matrices. In fact, all canonical forms for n x n(n≤ 5) upper trian-gular matrices with zero diagonal have been obtained. In this thesis we will study the canonical forms for n≥6 upper triangular matrices under upper triangular similarity. Applying Belitskii’s algorithm, we calculate the canonical forms of all the indecompos-able matrices under similarity for the linear matrix problem when n= 6,7. For n≥ 8, the lower bound of the indecomposable matrices under similarity for the linear matrix problem is obtained. Moreover, it is shown that it’s finite if n≤ 5; it’s tame if n= 6,7; it’s wild if n≥ 8.