前言

这个笔记是北大那位老师课程的学习笔记，讲的概念浅显易懂，非常有利于我们掌握基本的概念，从而掌握相关的技术。

Basic concepts

1. if ∑i=1,j=1naikijaj≥0\sum_{i=1,j=1}^{n}a_ik_{ij}a_j \geq 0 i=1,j=1∑n​ai​kij​aj​≥0∨a⊂R \vee a \subset R∨a⊂R
   then k is positive definite.
2. X∗Y⊂x∗yX*Y \subset x*yX∗Y⊂x∗y this is a set.
3. if kn−>k{k_n}->kkn​−>k then lim⁡x−>+∞aTkna≥0\lim\limits_{x->+\infin}a^Tk_na \geq 0x−>+∞lim​aTkn​a≥0
   =aTKa≥0=a^TKa \geq 0=aTKa≥0
   KK=>Kn−1k=knK^{n-1}k=k^nKn−1k=kn
   they all are P.S.D,then all ≥0\geq 0≥0
4. Thy: Let XXX be a nonempty set x0⊂Xx_0 \subset X x0​⊂X and Let ϕ:X∗X−>IR\phi :X*X ->I_Rϕ:X∗X−>IR​
   be a symmetric kernel.
   P−atK(x‾,y)=ϕ(x,x0)+ϕ(y,x0)−ϕ(x,y)=ϕ(x0,y0)P^-at K(\overline x,y)=\phi(x,x_0)+\phi(y,x_0)-\phi(x,y)=\phi(x_0,y_0)P−atK(x,y)=ϕ(x,x0​)+ϕ(y,x0​)−ϕ(x,y)=ϕ(x0​,y0​)
   then k is P.D iff ϕ\phiϕ is negative definite.