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Invertible Matrix #3

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chAwater opened this issue Feb 14, 2019 · 1 comment
Open

Invertible Matrix #3

chAwater opened this issue Feb 14, 2019 · 1 comment
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@chAwater
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chAwater commented Feb 14, 2019

Lecture 9: Linear Regression

线性回归求解时提到:

因为 N >> d+1 ,所以 的逆矩阵通常都是存在的。

为什么?

@chAwater chAwater added the help wanted Extra attention is needed label Feb 14, 2019
@chAwater
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从 Col 上的角度分析矩阵乘积:

因为 X 是一个 N x d+1 的矩阵,N >> d+1,所以 X 的 Col 是独立的;
的每个 Col 可以看成是 X 的每一个 Col 通过矩阵 得到的,所以 的每个 Col 也是独立的(否则与 X 的 Col 是独立的矛盾),所以逆矩阵存在。


或者用 Composition 的思路:

这个矩阵,可以看成是 X 两个线性系统的连接。

输入是一个 d+1 维的向量,先输入 X 这个系统,输出 N 维的向量;然后输入 这个系统,输出 d+1 维的向量。
因为 N >> d+1,从低维空间到高维空间的转化是可逆的,所以 这个系统是可逆的,所以逆矩阵存在。


呃,感觉很不严谨...甚至有可能错了....

@chAwater chAwater added enhancement New feature or request and removed help wanted Extra attention is needed labels Jul 28, 2020
chAwater added a commit that referenced this issue Jul 28, 2020
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