Eigenvector Slew - Solution

Solution

In terms of linear algebra this means that one wants to find an eigenvector with the eigenvalue = 1 for the linear mapping defined by

which relative to the

coordinate system has the matrix

\begin{bmatrix} \langle \hat{d}| \hat{a}\rangle & \langle\hat{e}| \hat{a}\rangle & \langle\hat{f}| \hat{a} \rangle \\ \langle\hat{d}| \hat{b}\rangle & \langle\hat{e}| \hat{b}\rangle & \langle\hat{f}| \hat{b}\rangle \\ \langle\hat{d}| \hat{c}\rangle & \langle\hat{e}| \hat{c}\rangle & \langle\hat{f}| \hat{c}\rangle \end{bmatrix}

Because this is the matrix of the rotation operator relative the base vector system the eigenvalue can be determined with the algorithm described in "Rotation operator (vector space)".

With the notations used here this is:

The rotation angle is

where "" is the polar argument of the vector corresponding to the function ATAN2(y,x) (or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN.

The resulting will be in the interval .

If then and the uniquely defined rotation (unit) vector is:

Note that

is the trace of the matrix defined by the orthogonal linear mapping and that the components of the "eigenvector" are fixed and constant during the rotation, i.e.

\hat{r}=r_x \cdot \hat{x}(t) +r_y \cdot \hat{y}(t) +r_z \cdot \hat{z}(t)= r_x \cdot \hat{a} +r_y \cdot \hat{b} +r_z \cdot \hat{c}= r_x \cdot \hat{d} +r_y \cdot \hat{e} +r_z \cdot \hat{f}

where are moving with time during the slew.

Read more about this topic:  Eigenvector Slew

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