Memo on ellipse and ellipsoid
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.
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.
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