My dad teaching Francisco to play chess (March 24, 2015).
HW 8 is due Tuesday, April 28, at the beginning of lecture.
Work in hyperbolic geometry. Given a triangle , recall that its Saccheri quadrilateral based at is defined as follows: Let be the midpoint of and be the midpoint of . Let be the feet of the perpendiculars from and to respectively.
Continuing with the same notation, suppose now that is an arbitrary point on , and let be a point on the ray with . Show that is also the Saccheri quadrilateral of based at .
I will be taking a leave from BSU this coming academic year, and moving to Ann Arbor, to work as an Associate Editor at MathReviews.
This exercise is due Tuesday, April 21, at the beginning of lecture.
Find the Singular Value Decomposition of
(I am not so interested in the specific answer, which can be found online, but rather in the process describing how one arrives to this answer.)
HW 6 is due Thursday, April 16 and HW 7 is due Tuesday, April 21, both at the beginning of lecture.
Work in hyperbolic geometry.
1. Let and be two parallel lines admitting a common perpendicular: There are points and with perpendicular to both and . Suppose that are other points in with , that is, is between and . Let be the foot of the perpendicular from to , and let be the foot of the perpendicular from to .
Show that . That is, and drift apart away from their common perpendicular.
(Note that and are Lambert quadrilaterals, and therefore and . The problem is to show that .)
As an extra credit problem, show that for any number we can find (on either side of ) such that , that is, and not just drift apart but they do so unboundedly.
2. Now suppose instead that and are critical (or limiting) parallel lines, that is, they are parallel, and if and is the foot of the perpendicular from to , then on one of the two sides determined by the line , any line through that forms with a smaller angle than does, cuts at some point.
On the same side as just described, suppose that are points on with , that is, is between and . Let be the foot of the perpendicular from to , and let be the foot of the perpendicular from to .
Show that . That is, and approach each other in the appropriate direction.
As an extra credit problem, show that for any we can choose so that . That is, and are asymptotically close to one another. Do they drift away unboundedly in the other direction?
Show that the critical function is continuous. Recall that measures the critical angle, that is, iff there are critical parallel lines and and a point such that if is the foot of the perpendicular from to , and , then and make an angle of measure in the appropriate direction.
(In lecture we verified that is strictly decreasing. This means that the only possible discontinuities of are jump discontinuities. We also verified that approaches as , and approaches as . It follows that to show that has no jump discontinuities, it suffices to verify that it takes all values between and , that is, one needs to prove that for any there is an such that .)
This optional homework is due Thursday, April 16, at the beginning of lecture. We want to verify the details of Jacobi method for diagonalizing symmetric matrices.
Recall that the Givens rotation has the effect of rotating a vector by radians in the -plane.
Suppose is an real symmetric matrix, and that . Let be the matrix where
- Show that if is chosen so that and , where , then .
- Show that .
- Show that .
This problem is due April 30 at the beginning of lecture.
Write a program that receives as input a real symmetric matrix and some tolerance bound , and performs the basic method to generating (and printing) a sequence of matrices until a stage is reached where the entries below the diagonal of are all in absolute value below . Once this happens, the program returns the diagonal entries of as approximations to the eigenvalues of . (Check on a couple of examples that these are indeed decent approximations, at least for of small size and reasonably small values of .)
Most Computer Algebra Systems already have implemented algorithms to find the decomposition of a matrix. Instead of using these pre-programmed algorithms, write your own.
(Turn in the code, plus the couple of examples. Comment your code, so it can be easily understood what you are doing along the way. I’m reasonably familiar with Maple, Mathlab, Sage, and most flavors of C. If you are going to use a different language, please let me know as soon as you can, to see whether it is something I’ll be able to verify or if a different language will be needed instead. Ideally, the user can choose the dimension of the input matrix.)