Normal form of the Frenet Matrix A book that I am going through states that on reducing to normal form the Frenet matrix in the Frenet-Serret formula, one ends up with the matrix: $$ K= \begin{bmatrix} 0 & \sqrt{\kappa^2+\tau^2} & 0 \\\ - \sqrt{\kappa^2+\tau^2} & 0 & 0 \\\ 0 & 0 & 0 \end{bmatrix} $$ $\kappa$ being the curvature and $\tau$ being the torsion of the curve in question. The term $$ \sqrt{\kappa^2 + \tau^2}$$ is called the angular velocity. I would like to know what Normal form they are alluding to? I am also confused as to what eigenvalues one gets from the Frenet formula matrix.

If you compute the characteristic polynomial of the Frenet matrix, you'll find that the eigenvalues are $0$ and $\pm i\sqrt{\kappa^2+\tau^2}$. Now, a real $2\times 2$ matrix $A$ with complex eigenvalues $\alpha\pm i\beta$ can be written in the form $$\begin{bmatrix} \alpha & \beta \\\ -\beta &\alpha\end{bmatrix}$$ by choosing as your basis the real and imaginary parts of a complex eigenvector. This is the normal form the text is referring to. (To verify this: Write $A(x+iy) = (\alpha+i\beta)(x+iy)$ and compare real and imaginary parts.)