In a paper I found in internet there is a relation about mapping a matrix into a vector and viceversa. One question has been already kindly answered here, but since I'm new to this kind of syntax I need more help.
In the paper on the third page the following relation is needed to calculate an error:
$ e_{R} = \frac{1}{2}(R_{d}^{T}R - R^{T}R_{d})^{\vee} $
which results to be a matrix. Now my concern is about the $\vee$ operator. Since it is supposed to map back a vector from that matrix, to be more precise $x \times y \mapsto y $ $\in \mathbb{R^{3}}$, I'm a little bit confused on the right way to proceed.
At first I thought I could map the matrix back to the sensor simply inverting the way I did it for getting the matrix from the vector. I mean, given the matrix $$ \begin{bmatrix} 0 & -c & b\\ c & 0 & -a\\ -b & a & 0 \end{bmatrix} $$ I simply copy that components into a vector: $x = [a, b, c]$ and that's all;
But reading more accurately the paper I found the following statement: "the $\vee$ map is the inverse of the hat map. We used the fact that $-\frac{1}{2}tr[\hat{x}\hat{y}] = x^{T}y$ for any $x,y \in \mathbb{R^{3}}$"... It suggests me that copying values into a vector is not a smart solution.
Now I'm a little bit confused on which way is better to get that vector from the resulting matrix:
$ \frac{1}{2}(R_{d}^{T}R - R^{T}R_{d}) = ? $
Thanks