Given , consider the ellipsoid
(a) Find the shortest and largest distance from the origin to the surface of the ellipsoid.
(b) Find the principal axes of the ellipsoid.
(c) Find an orthogonal transformation such that , thereby reducing the equation to the canonical form.
Solution. (c) Since
, so we have
. Now we try to find an orthogonal matrix R such that R'AR is a diagonal matrix. How to do this? We first need to find the eigenvalues

of A. We can set up the following equation.

.
We get three roots which are three eigenvalues of A as follows. ...

Solution Summary

This shows how to find the longest and shortest distances from origin to an ellipsoid, the principal axes, and an orthogonal transformation. The canonical form is examined.

2. A curve C has the parametrization x = a sint cos alpha, y = b sint sin alpha ,
z = c cost , t â?¥ 0 , where a, b , c, alpha are all positive constants.
a) Show that C lies on the ellipsoid x^2/a^2+y^2/b^2+z^2/c^2=1
b) Show that C also lies on a plane that contains the z axis.
c) Describe the curve C. Give its equatio

Let [EQUATION1] with [EQUATION2] and [EQUATION3]. The idea is to write each such set in some simple canonical form.
(i) When n = 2, how many distinct knapsack sets are there? Write them out in a canonical form with integral coefficients and 1 = [EQUATION4].
(ii) Repeat for n = 3 with [EQUATION5].
*(For proper equations an

For the given 4x4 matrix, find P such that INV(P)AP is in Jordan Canonical Form.
A = | 2 1 0 0 |
| -1 4 0 0 |
| 0 0 2 1 |
| 0 0 -1 2 |
It is easy to find repeated eigenvalues (3, 2+i, 2-i).
If I treat the upper block as a 2x2 matrix, I can find P = [1 0; 1 1] (Note: Using Matlab notation

Find the volume of the solid that is generated by rotating around the indicated axis the plane region bounded by the fiven curves.
1) y=√x,y=0,x=4; The x-axis
2) y= 1/x, y=0, x=0.1,x=1; the x-axis
3)Find the volume of the ellipsoid generated by rotating around the x-axis the region bounded by the ellipse with equation.

Central Conicoids (Part II)
Equation of the Tangent Plane to the Ellipsoid
Find the equation of the tangent plane to the ellipsoid 7x2 + 5y2 + 3z2 = 60 which pass through
the line 5y

1.
Which of the following mathematical relationships could be found in a linear programming model? And which could not (why)?
a. -1A + 2B < 70
b. 2A - 2B = 50
c. 1A - 2B2 < 10
d. 3 squareroot A + 2B > 15
e. 1A + 1B = 6
f. 2A + 5B + 1AB < 25
2.
Find the solutions that satisfy the following const