Project 5 Documentation
Math 2605, Gameiro
William Rorabaugh
In this project I took a given 2 x 2 matrix A and graphed the image of the unit circle
under A. From this graph (which I call the ellipse), I found the normalized vectors v1 and v2 that
pointed to the major and minor axis respectively of the ellipse. I also found the vectors u1 and u2
on the unit circle that mapped, respectively, to v1 and v2. I finally used the angles of u1 and v1,
and the lengths of the major and minor semi-axis to find the singular value decomposition of A.
I found v1 and v2 while I was finding the points on the ellipse from the points on circle
under A. In this loop, I tracked the vectors of the ellipse with the greatest and smallest
magnitudes. After the loop, I knew that these vectors were the major and minor semi axis and so
I normalized them to find v1 and v2. When I was tracking the largest and smallest vectors of the
ellipse, I also recorded the angle of the vector of the unit circle that I had just used to compute
each of them. I then used these angles I used to find v1 and v2 to derive the unit vectors u1 and u2.
After studying the symmetry of the ellipse and circle I noticed that if u1 maps to v1 then -
u1 maps to - v1. Using this fact, I decided that I could "flip" (multiply by -1) u1 and v1 into
specific quadrants to ease my later computations. So if v1 lay below the x-axis I flipped u1 and v1
to ensure that v1 was above the x-axis. Similarly, I flipped (if necessary) u2 and v2 into position
such that u2 lay clockwise of u1. Before going much further, I took the magnitude of the largest
and smallest vectors that I had found on the ellipse and reported them as the singular values of D
because the lengths of each these semi-axi are the product of the length of a unit vector and a
singular value.
After reading the confusing the instructions, I decided that matrix U should rotate or
reflect u1 and u2 onto (1, 0) and (0, 1) where matrix D would stretch them into the major and
minor axis. Since I had guaranteed that u2 was clockwise of u1, I knew that U would just need to
be a rotation. Since we would actually use UT in the SVD of A and since UT undoes the rotation
of U, I knew that the rotation angle of U was the angle of u1 (measured from + x axis) which
would be rotated back to 0 be UT. Now that I knew UT would rotate u1 and u2 to (1, 0) and (0, 1)
and that D would stretch (1, 0) and (0, 1) into the major and minor axis, I just needed to see how
V would rotate or reflect these axis into their final positions in the direction of v1 and v2.
Since I set v1 above the x-axis, I knew its y component was positive. Using a little
geometry, if the x component of v1 and the y component of v2 had the same sign, then I knew that
V did a simple rotation on the axis to position them at v1 and v2. Otherwise, v2 lay clockwise v1
and V must have done a reflection about the bisector of the angle between (1, 0) and v1. Thus, I
decided if V was an R rotation matrix or an M reflection matrix by comparing the signs of the x
component of v1 and the y component of v2. Since V rotated or reflected the major axis from the
direction (1, 0) to the direction v1, I knew that regardless if V was an R or an M, its angle was
the angle between (1, 0) and v1.
So after flipping all those little direction vectors and deriving the angles of U and V from
u1 and v1, and finally deciding whether V was a rotation or reflection, I had completed the SVD
of A that I could then displayed. The case of the degenerate ellipse (i.e. a line) didn't require any
special mathematics, although I didn't draw u2 and v2 if the length of v2 was almost zero. The one
special case that required some special mathematics was when the ellipses were circles. The
