一對圈型的線性變換

Abstract

若一個方陣X，其所有在對角線下方和最後一行第一列的項是非零，我們稱其為cyclic。令C代表一個體，V代表一個有限維佈於C的向量空間。我們稱一個在V上的cyclic pair，意思是一個有序對的線性變換A:V→V和B:V→V滿足下面(i), (ii)的條件。 (i)存在一組V的基底使A在此基底的矩陣表示法為對角矩陣和B在此基底的矩陣表示法為cyclic矩陣。 (ii)存在一組V的基底使B在此基底的矩陣表示法為對角矩陣和A在此基底的矩陣表示法為cyclic矩陣。 我們藉由他們矩陣係數和乘法運算規則來描繪cyclic pair。其中一個規則是和二項式定理相關。

A square matrix X is cyclic if all the entries in the lower diagonal and in the last column of the first row are nonzero. Let C denote a field and let V denote a vector space over C with finite positive dimension. By a cyclic pair on V we mean an ordered pair of linear transformations A:V→V and B:V→V that satisfies conditions (i), (ii) below. (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is cyclic. (ii) There exists a basis for V with respect to which the matrix representing B is diagonal and the matrix representing A is cyclic. We characterized cyclic pairs by their matrix coefficients, and by their multiplication rules. One of the rules is related to the binomial theorem.