摘要
矩阵的合同关系、相似关系都是等价关系 .它们虽然不同 ,但又有联系 .对称矩阵是这两个知识点的交汇点 ,即两个实对称矩阵合同当且仅当它们相似 .进一步得到二次型可以通过一个正交变换化为标准型 .这一理论是高等代数教科书的重要内容 .然而 ,现行的教科书对该理论的证明至少涉及到二次型、线性空间、线性变换和欧氏空间的内容 .本文利用欧氏空间的正交性质给出这一理论新的简洁证明 。
Both congruent and similarity of matrix are equivalence relations. The congruent and similar are different but relative. They cross at symmetric matrix, two symmetric matrices are congruent if and only if they are similar. Furthermore a quadric form can be changed to its standard form by orthogonal transformation. All tese are important contents in the advanced linear algebra. However the proof of this theory in existences is related to at least quadric form, linear space, linear transformation and Euclidean space. For the purpose of teaching, we give a new and explicit proof of this theory by using the orthogonal property of Euclidean space.
出处
《新疆大学学报（自然科学版）》
CAS
2004年第3期244-245,共2页
Journal of Xinjiang University(Natural Science Edition)
关键词
矩阵的合同
矩阵的相似
二次型
congruent of matrix
similarity of matrix
quadric form
