一對廣義圈型線性變換的研究

Abstract

令X是個方陣，當主對角線正下方及第一列最後一行的元素都非零，其他非對角線之元素皆為零，我們稱X為廣義圈型。在有限維的向量空間V中，如果兩線性變換A：V → V、B：V → V滿足下列條件(1),(2)則我們稱(A,B)為一對廣義圈型線性變換， (1) V中存在一個基底可以使的A的矩陣表示為對角矩陣，B的矩陣表示為廣義圈型。 (2) V中存在一個基底可以使的B的矩陣表示為對角矩陣，A的矩陣表示為廣義圈型。 我們將會給一對廣義圈型線性變換存在的兩個必要條件。

Let X be a square matrix. We say X is weak cyclic when each of the entries in the lower diagonal and in the last column of the lower diagonal are nonzero and all the other nondiagonal entries of X are zero. Let V denote a vector space over C with finite positive dimension. By a weakly 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 weakly cyclic. (ii). There exists a basis for V with respect to which the matrix representing B is diagonal and the matrix representing A is weakly cyclic. We give two necessary conditions among the eigenvalues and the coefficients in some representing matrix of a weak cyclic pair.