# 7.3.2 正定核

这一节大部分没看懂。就把看懂的部分记一下。

## 证明：Gram矩阵是半正定的，则存在从X到H的映射$$\phi$$

公式7.69：

$$
\phi:x \rightarrow K(\cdot,x)
$$

这是一个映射函数，把一个向量x映射成另一个向量$$K(\cdot,x)$$

公式7.70：

$$
f(\cdot) = \sum\_{i=1}^ma\_iK(\cdot, x\_i)
$$

$$x\_i$$是x中的任意向量。$$f(\cdot)$$是x是任意向量的[线性组合](https://windmising.gitbook.io/mathematics-basic-for-ml/xian-xing-dai-shu/vector)。

集合S：$$K(\cdot, x\_i)$$的各种线性组合的结果构成一个集合。

S构成[向量空间](https://windmising.gitbook.io/mathematics-basic-for-ml/xian-xing-dai-shu/vector)：因为S满足加法封闭性的乘法封闭性。

S构成[内积空间](https://windmising.gitbook.io/mathematics-basic-for-ml/xian-xing-dai-shu/vector)： 具有内积运算的向量空间是内积空间。因此要为S定义一个满足内积属性的内积操作。\
令A，B是S上的两个元素，分别是$$K(\cdot,x)$$的两种任意的线性组合的结果。定义S上的内积操作为：

$$
\begin{aligned}
A = \sum\_{i=1}^ma\_iK(\cdot, x\_i)   \\
B = \sum\_{j=1}^l\beta\_jK(\cdot, x\_j)  \\
A \cdot B = \sum\_{i=1}^m\sum\_{j=1}^la\_i\beta\_jK(x\_i, z\_j)
\end{aligned}
$$

这个内积操作满足属性：对称性、左线性、正定性

**从3开始就看不懂了**

## 证明：K(x,z)是正定核的充要条件是K(x,z)对应的Gram矩阵是正定的。


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://windmising.gitbook.io/lihang-tongjixuexifangfa/11/14.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.