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Memo on ellipse and ellipsoid


Edit: 2015-10-05

Useful fact: if A\ge B, then \sigma_{\min}(A)\ge \sigma_{\min}(B) and \sigma_{\max}(A)\ge \sigma_{\max}(B)

Proof: For the eigenvector x1 that associated with the smallest eigenvalue of A

\sigma_{min}(A)=x_1^TAx_1\ge x_1^TBx_1\ge \sigma_{min}(B)

For the eigenvector yn that associated with the largest eigenvalue of B

\sigma_{max}(B)=y_n^T B y_n\le y_n^TAy_n\le \sigma_{max}(A)

Remark: it is not correct that \sigma_{min}(A)\ge \sigma_{max}(B). Counterexamples can be easily found if you consider the geometric interpretation of the ellipsoids x^T*A*x=1 and x^T*B*x=1.


Edit: 2015-09-23

Fact 1.
If x^TAx=1, then it defines an ellipsoid.
If x^TAx<1, then the vector x is inside the ellipsoid.
If x^TAx>1, then x is outside the ellipsoid.

Proof: suppose x^T Ax<1. Then there exists k^2=1/(x^TAx) such that (kx)^TA(kx)=1. Therefore, kx is on the surface of the ellipsoid and since k>1, we know x is inside the ellipsoid.

Fact 2. For any PD matrices A and B, if A>B (i.e., A-B>0), then x^TAx=1 is contained inside of x^TBx=1.

Remark: The more positive A is, the smaller the ellipsoid it is. Roughly speaking, that is because the length of each semi-axis is the inverse of the eigenvalue.
Remark: note A and B may have different eigenvectors. So if you plot the two ellipsoids, they are not parallel.
Remark: you cannot naively say that the singular values of A are larger than those of B even if A>B. This can be intuitively seen by plotting the two ellipsoid. But we can say the largest (smallest) eigenvalue of A is larger than the largest (smallest) one of B. This can be seen from the plot and also proved. We cannot say the smallest eigenvalue of A is larger than then maximum eigenvalue.

Proof: If x^TAx=1, then x is on the surface of the ellipsoid of A. Then x^TBx=x^TAx-x^T(A-B)x<x^TAx=1. Hence x is inside of the ellipsoid of B. Hence any point on the ellipsoid of A is inside the ellipsoid of B.


x^T A x=1 represents an ellipsoid.

  1. The axis of the ellipsoid is along the eigenvectors of A.
  2. The length of the semi-axis is 1/\sqrt{\lambda_i}.
  3. The volume of the ellipsoid is \frac{4}{3}\pi \frac{1}{\sqrt{\lambda_1 \lambda_2 \lambda_3}}.

Here is a detailed memo. memo – ellipsoid

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