一些英文的数学名词

Normal Vector

The normal vector, often simply called the "normal," to a surface is a vector perpendicular to it. Often, the normal unit vector is desired, which is sometimes known as the "unit normal." The terms "normal vector" and "normalized vector" should not be confused, especially since unit norm vectors might be called "normalized normal vectors" without redundancy. When normals are considered on closed surfaces, the inward-pointing normal (pointing towards the interior of the surface) and outward-pointing normal are usually distinguished. The normal vector is commonly denoted  or , with a hat sometimes (but not always) added (i.e.,  and ) to explicitly indicate a unit normal vector.

Projection matrix(投影矩阵)

$P = A(A^TA)^{-1}A^T$ is a projection matrix where $A = [1,m]^T$

Property:

$\cdot$ $P$ is called a projection matrix if $P^2=P$

$\cdot$ $I$ is a projection matrix

Convex set and Convex Function

证明向量空间是否为凸集的方法：

假设X,Y在空间中，则有任意$a(0<= a <= 1)$使得$aX+(1-a)Y$仍然属于这个向量空间，那么这个向量空间及时凸集。

凸函数：

一个定义在某个向量空间的凸子集$C$上的实值函数$f$,如果在定义域$C$上的任意两点$x,y$，以及$t\in [0,1]$,有

\[f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)\]

也就是说，一个函数是凸的当且仅当其上镜图（在函数图像上方的点集）为一个凸集。

Matrix theory

column vector $p_i$,$q_i$, and a constant column vector $y_k$.

\[p_i^T(q_i-y_k)=(q_i-y_k)^Tp_i\]

向量积

向量积也被称为矢量积、叉积，是一种在向量空间中向量的二元运算。与点积不同，他的运算结果是一个向量而不是一个标量。并且两个向量的叉积与这两个向量垂直。

$a\times b=(a_yb_z-a_zb_y)i+(a_zb_x-a_xb_z)j+(a_xb_y-a_yb_x)k$，为了帮助记忆，利用三阶行列式，写成

\[[[i,j,k];[a_x,a_y,a_z];[b_x,b_y,b_z]]\]