An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid *equivalent* if one can be obtained from the other by rotation of space. 


Now suppose that $E$ and $E'$ are two equivalent centric ellipsoids that are both contained inside a third centric ellipsoid $F$. Is it always possible to continuously rotate $E$ inside $F$ until is gets to the position $E'$? 


This question is a more difficult version of [this one](https://mathoverflow.net/questions/107483).



Can an ellipsoid be moved freely inside another ellipsoid?
The answer is yes.


You can connect your small ellipsoid to the big one by a continuous nested one-parameter family of ellipsoids, say $E\_t$, $t\in [0,1]$.
Nested means that $E\_{t\_1}\subset E\_{t\_2}$ if $t\_1 \le t\_2$. 
Say if 
$$E=\{\, x \mid \langle x,Ax\rangle \le1 \,\}\ \ \text{and}\ \ F=\{\, x \mid \langle x,Bx\rangle \le1 \,\}$$ 
then 
$$E\_t=\{\, x \mid \langle x,((1-t)\cdot A+t\cdot B)x\rangle \le1 \,\}$$


Let $E$ and $E'$ be two ellipsoids with semiaxis correspondingly $a\_1\le a\_2\le\dots\le a\_n$
and $a'\_1\le a'\_2\le \dots \le a'\_n$. 
Note that if $E\supset E'$,
then $a\_i\ge a\_i'$ for all $i$.


Now consider continuous family of rotations of $E=E\_0$ such that its axis go to the axis of $E\_t$.
This way we get a continuous rotation of $E\_0$ inside $E\_1=F$ which moves it to the standard position.
One has to be bit more careful in case some of $E\_t$ have equal semiaxis.
The later is left as an exercise.


