[Math] Relationship between ellipsoid radii and eigenvalues

conic sectionseigenvalues-eigenvectorsellipsoids

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup.

I went through a theorem in the book stating the relationship between ellipsoid's radii and the eigenvalues.

So the ellipsoid is defined as following:

$\sum_i \sum_j (x_i – u_i)(x_j-u_j)c_{ij} \leq k$

where $C=[c_ij]$ is an symmetric positive definite matrix, $x, u$ are n-dimensional vectors. The book says that the radii of this ellipsoid is $r_i = \frac{k}{\sqrt{\lambda_i}}$ where $\lambda_i$ are the eigenvalues of matrix C. However it does not give any proof for that. I've tried to google for the proof but didn't get anything useful.
Could someone please help me understand it?

Thanks,

Best Answer

You can start from the disc representation and gradually make the examples more complicated. Example: Let $p$ a point in the two dimensional plane $$ x^2+y^2 \leq 1 $$ can be rewritten as $$ \pmatrix{x&y}\pmatrix{1&0\\0&1}\pmatrix{x\\y} \leq 1 $$ This is the simplest case where you can see the resulting shape and read off the eigenvalues at the same time. Smashing the circle from the top and bottom would result with $$ \pmatrix{x&y}\pmatrix{1&0\\0&4}\pmatrix{x\\y} \leq 1 $$ We can see that points satisfy this can at most have a $y$ component with magnitude $0.5$ etc. We can further generalize this to circles that is centered somewhere other than the origin and also we can rotate these eccentric ellipsoids. When rotated, instead of plain $x$ and $y$ axes being principal axes you get the eigenvectors working for you and the lengths are associated with the magnitude of the eigenvalues. You can show this by eigenvalue decomposition or simply let your variables $x,y$ be the eigenvector of the largest eigenvalue and see them as rotated $(x,y)$ pairs that we have started with.

In general, we are left with inequalities in the form of $$ \pmatrix{p-p_0}C\pmatrix{p-p_0} \leq s $$ where $C$ is a positive definite matrix, $p_0$ is the center of the ellipsoid and $s$ is, roughly, the size of the ellipsoid. The positive definiteness is not crucial but often physically meaningful. If the matrix is not positive definite you allow for the reflections along some axis.

Related Solutions

[Math] Plotting an ellipsoid using eigenvectors and eigenvalues

Turns out I was indeed doing two things wrong!

1. Axes order
Like I suspected — and mentioned in the OP as an important note — the frame defined by the axes of unit eigenvectors $x$ was sometimes switching direction, and thus not respecting the right-hand rule and world frame convention.

To overcome this, I just reassigned the 3rd eigenvector as the cross product between the 1st and 2nd eigenvectors:

x[:, 2] = np.cross(x[:, 0], x[:, 1])

Since the eigenvectors $x$ returned by numpy.linalg.eig are already normalized (i.e., unit eigenvectors), there is no need to normalise the result of the cross product.

2. Rotation matrix
The second thing I was not doing properly was during the construction of the $3$x$3$ rotation matrix. I was missing a transpose operation.

To solve this I just changed the indexing of the eigenvectors $x$ to:

rot_matrix = np.array([x[0, :],
                       x[1, :],
                       x[2, :]])

Using numpy.transpose would have worked too.

Below is the correct expected final result:

final-result

Special thanks to Vladimir Ivan (a.k.a. "Slovak")!

[Math] The relationship between spectral decomposition / eigendecomposition and projection operators

Does this mean, then, that the projection operator associated with λi can be related to the sum of outer products of eigenvectors with the same eigenvalue:

Yes. For a normal matrix, when $T$ is diagonalizable, it can be decomposed into: $$ T = \lambda _1P_1 + \lambda _2P_2 + ... $$

The Projection matrices $P_i$ or $q_j q_j^*$ form eigenspaces. For a repeated eigenvalue, the corresponding eigenvectors form the basis of an eigenspace. These eigenspaces are orthogonal to each other.

for instance, I believe that V needs to be an inner product space for the spectral decomposition to be formulated)

Spectral Theorem states that Every Normal matrix can be diagonalized by a complete set of orthonormal eigenvectors. So I don't think any other condition needs to be satisfied other than the normality condition.

Best Answer

Related Solutions

[Math] Plotting an ellipsoid using eigenvectors and eigenvalues

[Math] The relationship between spectral decomposition / eigendecomposition and projection operators

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