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I have an ellipsoid defined by the semiaxes $a,b,c$ and the orthonormal vectors $v_x, v_y, v_z$ describing the directions in which the axes point. The matrix $\Lambda=\left(\begin{matrix}a&0&0\\0&b&0\\0&0&c\end{matrix}\right)$ then simply represents the ellipsoid in its "native" coordinate system while $A=T\cdot\Lambda$ with $T=\left(v_x,v_y,v_z\right)$ represents the ellipsoid in the "laboratory" coordinate system in which it is arbitrary oriented.

Now: Shouldn't the Eigenvalues of $\Lambda$ and $A$ be the same and be related to the shape of the ellipsoid irrespective of the ellipsoid orientation given by $T$, i.e. $EigenValues(A)=(a,b,c)$?

When i create the rotation matrix $T=\left(\begin{matrix}Cos(\Theta)&Sin(\Theta)&\\Sin(\Theta)&Cos(\Theta)&0\\0&0&1\end{matrix}\right)$ and calculate the Eigenvalues of $T \cdot \Lambda$ for different values of $\Theta$ with mathematica, i get varying results.

Angle, Eigenvalue1, Eigenvalue2, Eigenvalue3 0 3, 2, 1 45°, 3, 1.06+0.93j, 1.06-0.93j 180°, 3, 0+1.41j, 0-1.41j 275°, 3, -1.06+0.93j, -1.06-0.93j Where's my mistake?

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  • $\begingroup$ I am not sure if I understand well. The eigenvalues give you the length of principal axes so no matter the orientation of your ellipsoid, they should remain the same. As for the orientation you just need $n$ orthogonal vectors. All you need is to use the similarity transformation of one matrix to another. $\endgroup$ – Jan Jul 7 '14 at 12:58
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You've forgotten to multiply $T\cdot\Lambda $ by $T^{-1}$ on the right - when we change bases/coordinates we have to make sure that our linear map is expressing a transformation in this new basis. The $T^{-1}$ above ensures this description is complete. The resulting matrix $T\cdot\Lambda \cdot T^{-1}$is then $similar$ to our original matrix $\Lambda$. As similar matrices represent the same linear map just with respect to different bases, we should expect that they respect qualities of linear maps that are invariant under a change of basis, i.e. eigenvalues, dimensions of eigenspaces, the characteristic polynomial, determinant and trace to name but a few.

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