Surjective group homomorphism

[e12514]

Homework Statement

Show the map (call it phi) from U_n to C*
defined by phi(X) = det(X) for all matrices X in U_n,
is a surjective homomorphism, where

U_n is the subgroup of GL(n,C) consisting of unitary matrices
C* = C\{0} = invertible/nonzero complex numbers
det(.) is the determinant of .

Homework Equations

A matrix X in GL(n,C) is unitary if ((X_bar)^T).X = I
where X_bar is the conjugate of X (taking the conjugate of each entry in X)

The Attempt at a Solution

Homomorphism is easy to verify:
phi(XY) = det(XY) = det(X) det(Y) = phi(X) phi(Y)


I'm having trouble showing it's surjective.

For it to be surjective we need
for all (a+bi) in C* there exists X in U_n such that phi(X) = det(X) = a+bi
where a and b not both zero

The problem is X needs to be unitary, that is, X-conjugate-transpose times X needs to be the identity matrix.
It follows the the required matrix X can't be a diagonal matrix, otherwise it is not unitary if its determinant is (a+bi), in particular every entry on that diagonal where there's a complex number x+yi, that entry will become (x^2 + y^2) after taking ((X_bar)^T).X instead of the required "1" on the diagonal of an identity matrix.

Similarly I couldn't get anywhere with triangular matrices, and any non-diagonal and non-triangular matrix seems to get too complicated, not to mention that triangular matrices are messy enough...

 

[morphism]

It's definitely not going to be surjective -- the determinant of a unitary matrix lies on the unit circle.

 

[e12514]

Hmm... that's true. Maybe there was a typo on the question.
Unless there is some other interpretation for what C* is...

 

[CompuChip]

Maybe it was indeed meant that
C* = { z \in C : ||z|| = 1 }