2. What is the geometric meaning of a vector multiplying its transpose?

Reference resources ：https://www.zhihu.com/question/40049682/answer/1420483558

There are two situations ：

One 、 That's ok X Column

Is the square of its length .

Two 、 Column X That's ok

  It is usually dealt with （ normalization ）：

  For any vector   b , It projects to   a   The vector on must be ：

 --------------------------------------------------------------------------------------------------------------------------------

About columns X That's ok Proof of projection of ：

 

 

There are two vectors , hold stay Projection notation of .

This process is obviously a linear transformation , So let's record this linear transformation as ： .

So according to its definition, there are ：

  here It's scalar . that Nature is represented in the vector The vector over .

Consider from the triangle rule

vector It can be seen as The error of the .

It is not difficult to see from the geometric relationship  

 

 

  Then it will be transformed into a quantitative relationship

It is not difficult to draw a conclusion from the equivalence relation of the marked red ：

Note that the default vectors are column vectors , So scalars can be written directly as ：

At this point, let's turn back to linear transformation

That's what's mentioned above Projection matrix The whole process of proof .