how to compute norm of a vector of dual numbers?

Question

My Question is,

How to compute norm of a vector of dual numbers? Is is same as complex number because the dot product is the same with the difference of $\epsilon^2 = 0$ as opposed to $i^2 = -1$.

Answer 1

If you define the norm of a dual number $z=a+b\varepsilon$ by $|z|^2 = z\overline{z}$ where $\overline{z} = a-b\varepsilon$ you get $$ |z|^2 = (a+b\varepsilon)(a-b\varepsilon) = a^2-b^2\varepsilon^2 = a^2 $$ so that $|z|=|a|$. Note however, that this is not a "norm" is the sense that $|z|=0$ does not imply $z=0$.

Now you can do the same for vectors of dual numbers by letting $\langle u, v\rangle = u^T \overline{w}$ but you won't get a positive definite product and hence not a norm.