Matrix Projection
最近在看一些有關data-driven dynamic systems e.g. DMD 時常用到矩陣投影的觀念. 例如：Ｍatrix B project onto space spanned by column of A. 所以上網找了些資料，在這邊做了整理並記錄。 一直以來，我認為要用幾何的觀點來看matrix的運算. Let's star with vector projection. The figure shows the project vector $b$ onto the space space spanned by matrix column vector $a_1$ and $a_2$. Assume the vector $p$ spanned by $a_1$ and $a_2$ which are linearly independent, $$p=x_1a_1+x_2a_2=ax $$. The key point is error vector $e=b-p$ 垂直於 $a$. That is 內積 $$a^T(b-p)=0$$. Therefore, $x=\frac{a^T b}{a^Ta}$, $p=xa=\frac{a^T b}{a^Ta}a$ Define Projection matrix $$P=\frac{aa^T}{a^Ta}$$ which projects any vector $b$ onto $a$, $p=Pb=\frac{aa^T}{a^Ta}b$.