一类二阶整数矩阵方程的解  

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

作　　者：尹倩倩 梁欣然 袁平之[1] YIN Qianqian;LIANG Xinran;YUAN Pingzhi(School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China)

机构地区：[1]华南师范大学数学科学学院

出　　处：《华南师范大学学报（自然科学版）》2019年第5期104-109,共6页Journal of South China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11671153);广东省自然科学基金项目(2016A030313850)

摘　　要：为解决与毕达哥拉斯方程x^2+y^2=z^2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X^2+Y^2=λI(λ∈Z,I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X^2-Y^2=λI的全部解.To solve problems of integer matrix equations related to Pythagorean equation x^2+y^2=z^2,the solutions(X,Y)to the 2×2 integral matrix equation X^2+Y^2=λI,where λ∈Z 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 X^2-Y^2=λI also can be solved with similar methods.

关 键 词：二阶整数矩阵 矩阵方程 不定方程 毕达哥拉斯方程 特征值 上三角化 

分 类 号：O156.7[理学—数学]