# Relationship between Trace and Determinant of Unitary Matrices

1. Mar 24, 2012

### dpeagler

1. The problem statement, all variables and given/known data

If U is a 2 x 2 unitary matrix with detU=1. Show that |TrU|≤2. Write down the explicit form ofU when TrU=±2

2. Relevant equations

Not aware of any particular equations other than the definition of the determinant and trace.

3. The attempt at a solution

I have attempted this problem in several different ways to no avail. I'm pretty sure that the explicit form of U when the trace is 2 is simply the 2 x 2 identity matrix.

I do know that

Tr(AB) ≠ Tr(A)Tr(B)

and

U*U = I

but not sure where to go with this.

Any help is greatly appreciated.

Last edited: Mar 24, 2012
2. Mar 24, 2012

### sgd37

remember that for diagonal matrices the determinant is given by the product of the eigenvalues

3. Mar 24, 2012

### I like Serena

Perhaps you can use one of the equivalent definitions of a unitary matrix?
See: http://en.wikipedia.org/wiki/Unitary_matrix
I'm thinking that the columns of a unitary matrix form an orthonormal basis...

4. Mar 25, 2012

### dpeagler

Sgd37 I am aware that the product of the eigenvalues has to be 1, but there are an infinite number of combinations that I can create. I'm not sure if the eigenvalues have to be real or not since I don't think that the unitary matrix is necessarily Hermitian.

I'm probably missing something simple.

Thanks so much.

5. Mar 25, 2012

### I like Serena

In an orthonormal basis, the vectors have length 1.
So the highest value in such a vector is 1.

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