﻿ 高等代数与数学分析相互渗透溶合应用研究 Research on Mutual Application of Advanced Algebra and Mathematical Analysis Theory

Vol.06 No.03(2017), Article ID:20587,5 pages
10.12677/AAM.2017.63028

Research on Mutual Application of Advanced Algebra and Mathematical Analysis Theory

Liu Yang, Chengxin Luo

College of Mathematics and System Science, Shenyang Normal University, Shenyang Liaoning

Received: Apr. 28th, 2017; accepted: May 13th, 2017; published: May 22nd, 2017

ABSTRACT

Theory in mathematical analysis is different from that in advanced algebra and related problems in higher algebra. The problems discussed and the methods to solve the problems are different, but there is a close relationship between them. This paper illustrates the mutual penetration and integration of mathematical analysis and related problems in higher algebra by some examples, we present a method to solve algebra problem by functional analysis theory.

Keywords:Symmetric Matrix, Orthogonal Transformation, Compression Mapping Principle, System of Linear Equations

1. 引言

2. 预备知识

3. 用高等代数方法解决数学分析问题

，从而，

.

, (1)

，而

4. 用数学分析方法解决高等代数问题

(2)

，则上面的广义积分中被积函数等于。对于任意不为零的向量亦不为零。而为正定矩阵，故二次型正定，即对于任意恒成立。综上所述，任意都有，则，被积函数大于零，积分值一定大于零。所以为正定二次型，为正定矩阵，证完。

5. 数学分析与高等代数相结合解决问题

(2)

(代数法) 令，设个特征值，令，即，其中的属于的正交单位向量；，则，显然

,

6. 用泛函分析法处理代数问题

Research on Mutual Application of Advanced Algebra and Mathematical Analysis Theory[J]. 应用数学进展, 2017, 06(03): 238-242. http://dx.doi.org/10.12677/AAM.2017.63028

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