# Memo on ellipse and ellipsoid

Edit: 2015-10-05

**Useful fact**: if , then and

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

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

**Remark**: it is not correct that . 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.

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Edit: 2015-09-23

**Fact 1.
**If , then it defines an ellipsoid.

If , then the vector x is inside the ellipsoid.

If , then x is outside the ellipsoid.

**Proof**: suppose . Then there exists such that . Therefore, is on the surface of the ellipsoid and since , we know is inside the ellipsoid.

**Fact 2.** For any PD matrices A and B, if A>B (i.e., A-B>0), then is contained inside of .

**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 , then x is on the surface of the ellipsoid of A. Then . Hence x is inside of the ellipsoid of B. Hence any point on the ellipsoid of A is inside the ellipsoid of B.

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represents an ellipsoid.

- The axis of the ellipsoid is along the eigenvectors of A.
- The length of the semi-axis is .
- The volume of the ellipsoid is .

Here is a detailed memo. memo – ellipsoid