今天做作业的时候,碰到有关正交矩阵的题目,有点不是那么懂原理那些,就找下资料学一下,现在就将我所学到的知识点,列出来,供大家学习参考。如果有哪里写的不对的话,就留言告诉博主一下!若觉得写的还不错的话,就点个赞吧!
先看下怎么定义正交矩阵吧:
A matrix A is orthogonal if A T A = I.
这个定义是不是很难读的懂啊,没关系,给各位一个例子看就知道了。
那么,矩阵A就是正交矩阵。
讲完定义了,就接着看下它有什么属性,这对以后做题很有帮助。
属性 1:If A is an m x n orthogonal matrix and B is an n x p orthogonal, then AB is orthogonal.
证明:(AB)T (AB) = (BT AT )(AB) = BT (AT A)B = BT I B = BT B = I
属性 2: If A is an orthogonal square matrix, then AT = A-1
属性 3:If A is an orthogonal square matrix, then AAT = I
属性 4:If A is an orthogonal square matrix, then AT is orthogonal
证明:(AT )T AT = AAT = I
属性5:If A is an orthogonal square matrix, then det A =
1
证明:|A|2 = |A|
|A| = |AT |
|A| = |AT A| = |I| = 1. Thus, |A| =
1.