My CAS HP Prime calculator gives only unit vectors when I use the function for Eigenvectors. I know how to find a unit vector, $u=\frac{v}{||v||}$, but that doesn't seem to help.
For example:
Eigenvector$\begin{pmatrix}1.25&0.75\\ 0.75&1.25\end{pmatrix}\rightarrow P=\begin{pmatrix}\frac{\sqrt{2}}{2}&-\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{pmatrix}, D=\begin{pmatrix}2&0\\ 0&0.5\end{pmatrix}$
Is there a quick trick to get the full vector $\begin{pmatrix}1&-1\\ 1&1\end{pmatrix}$ instead of a unit vector? (Aside from multiplying P by $\frac{2}{\sqrt{2}}$).
A trick that can be applied to something like this too, to derive the full vector instead of unit vector, my calculator gives the following:
Eigenvector $\begin{pmatrix}8&12&-18\\ 4&18&-20\\ 4&13&-15\end{pmatrix} \rightarrow P= \begin{pmatrix} \frac{849027}{1166876} & \frac{1013883}{347759} & \frac{3880899}{1372105}\\ \frac{283009}{583438} & \frac{1013883}{347759} & \frac{7095941}{1254397}\\\frac{283099}{583438} & \frac{1013883}{347759} & \frac{5488420}{1293633}\end{pmatrix}, D=\begin{pmatrix}4&0&0\\ 0&2&0\\ 0&0&5\end{pmatrix}$ .
Want to get P = $\begin{pmatrix}3&1&2\\ 2&1&4\\ 2&1&3\end{pmatrix}$
Do something like Euclid's algorithm:
Take the difference between terms, and difference between differences. When a difference is much much smaller, assume it is zero.
Take the smallest of the reasonable-sized differences as your unit, and divide each of the original terms by that. Then round to the nearest integers.
A faster method is to simply divide the vector by its smallest nonzero entry, and round to the nearest simple rational numbers.