We defined a norm as:
\(||v||=v^TMv\)
A metric is the distance between two vectors.
\(d(u,v)=||u-v||=(u-v)^TM(u-v)\)
A set with a metric is a metric space.
Metric spaces can be used to induce a topology.
The distance between two vectors is:
\((v-w)^TM(v-w)\)
So what operations can we do now?
As before, we can do the transformations which preserve \(u^TMv\), such as the orthogonal group.
But we can also do other translations
\((v-w)^TM(v-w)\)
\(v^TMv+w^TMw-v^TMw-w^TMv\)
so symmetry is now \(O(3,1)\) and affine translations
\([[1,x][0, 1]]\) moves vector by \(x\).
This generalises the Euclidian norm.
\(||x||_p=(\sum_{i=1}^{n}|x|^p_i)^{1/p}\)
This can defined for different values of \(p\). Note that the absolute value of each element in the vector is used.
Note also that:
\(||x||_2\)
Is the Euclidian norm.
This is the \(L^1\) norm. That is:
\(||x||_1=\sum_{i=1}^{n}|x|_i\)
We can use norms to denote the "length" of a single vector.
\(||v||=\sqrt {\langle v, v\rangle }\)
\(||v||=\sqrt {v^*Mv}\)
If \(M=I\) we have the Euclidian norm.
\(||v||=\sqrt{v^*v}\)
If we are using the real field this is:
\(||v||=\sqrt{\sum_{i=1}^{n}v^2_i}\)
If \(n=2\) we have in the real field we have:
\(||v||=\sqrt{v_1^2+v_2^2}\)
We call the two inputs \(x\) and \(y\), and the length \(z\).
\(z=\sqrt {x^2+y^2}\)
\(z^2=x^2+y^2\)
This states that:
\(|\langle u,v\rangle |^2 \le \langle u, u\rangle \dot \langle v, v\rangle\)
Or:
\(\langle v,u\rangle\langle u,v\rangle \le \langle u, u\rangle \dot \langle v, v\rangle\)
\(\langle v,u\rangle\langle u,v\rangle \le \langle u, u\rangle \dot \langle v, v\rangle\)
\(\dfrac{\langle v,u\rangle\langle u,v\rangle}{||u||.||v||} \le ||u||.||v||\)
\(\dfrac{||u||.||v||}{\langle v,u\rangle} \ge \dfrac{\langle u,v\rangle}{||u||.||v||}\)
\(\cos (\theta )=\dfrac{\langle u,v\rangle }{||u||.||v||}\)