Recall that the upper half plane
This assignment is due Friday, October 18, 2013 at the start of class.
denote the collection of those real matrices with nonzero determinant. Prove the Iwasawa Decomposition:
in terms of the real numbers
, , , and .
denote those real matrices with positive determinant. Consider the map defined by . Show that is surjective, whereas .
Problem 2. Show that
Problem 4. Let
- Identifying the affine complex line
with the affine real plane , show that is a circle of radius centered at which intersects the boundary at the two distinct complex points . Hint: Solve the simultaneous equations .
- Show that
is perpendicular to at the points where they intersect. Hint: Consider the inner product of the vectors and which are normal to and , respectively.
Problem 5.Using the isomorphism
show that the intersection of the unit disk