一类二阶整数矩阵方程的解

Solutions to a Class of Equations in Integral Matrices of Order 2

摘要: 为解决与毕达哥拉斯方程x²+y²=z²相关的整数矩阵方程问题, 利用矩阵的基本运算把整数矩阵方程问题转化成不定方程求解的问题, 从特殊情形逐步推广到一般情形, 研究了与毕达哥拉斯方程相关的一类二阶整数矩阵方程\mathit\boldsymbolX^2 + \mathit\boldsymbolY^2 = \lambda \mathit\boldsymbolI (\lambda \in \mathbbZ, \boldsymbolI 为单位矩阵), 并得到其全部解( X , Y ), 类似可得二阶整数矩阵方程\mathit\boldsymbolX^2 - \mathit\boldsymbolY^2 = \lambda \mathit\boldsymbolI 的全部解.

Abstract: To solve problems of integer matrix equations related to Pythagorean equation x²+y²=z², the solutions ( X , Y ) to the 2×2 integral matrix equation \mathit\boldsymbolX^2 + \mathit\boldsymbolY^2 = \lambda \mathit\boldsymbolI , where \lambda \in \mathbbZ and I is the unit matrix, which are related to the Pythagorean equation, are investigated and completely solved by using the basic operation of matrix to transform the problem of integer matrix equation into the problem of solving some Diophantine equations, which is gradually extended from the special case to the general case. The solutions to 2×2 integral matrix equation \mathit\boldsymbolX^2 - \mathit\boldsymbolY^2 = \lambda \mathit\boldsymbolI also can be solved with similar methods.