Proving Determinant of Unitary Matrix is Complex Number of Unit Modulus

[skrtic]

SOLVED









1. show that the determinant of a unitary matrix is a complex number of unit modulus







2. i know the equation for a determinant, but i guess to i am not sure what a complex number of unit modulus is either. I'm looking for guidance

 

[gabbagabbahey]

but i guess to i am not sure what a complex number of unit modulus is either. I'm looking for guidance

The modulus of a complex number $z=x+iy$, where $x$ and $y$ are real numbers representing the projections of $z$ onto the real and imaginary axes respectfully, is simply given by $|z|=\sqrt{x^2+y^2}$.

So a complex number with unit modulus is simply a complex number $z$ such that $|z|=\sqrt{x^2+y^2}=1$.

To find the determinant of a unitary matrix, start with the definition of unitary matrices (in the form of an equation) and take the determinant of both sides of the equation.

 

[skrtic]

well my problem gives the matrix of [[a,b][c,d]] and gives the det([[a,b][c,d]])=ad-bc

then states the question i gave above.

i read that the |det(unitary matrix)|=1, but isn't that what i am trying to solve for.

and i am not sure if i have seen the definition of unitary matrices in the form of an equation.

right now this is for a high level undergrad quantum course which i have to take self paced and this is my first hurdle.

 

[skrtic]

this is my attempt i just thought about. call the matrix R

abs(det(R x R*))=1 since R x R* is I and det(I) = 1

and then abs(det(R) x det(R*))=1

and i get to a^2d^2+b^2c^2=1

but i don't know if that does anything for me

 

[gabbagabbahey]

this is my attempt i just thought about. call the matrix R

abs(det(R x R*))=1

You seem to be starting with the result you are trying to prove...looks like circular logic to me...

since R x R* is I

This is the definition of a unitary matrix, and this is what you should start with.


So start with $$RR^{*}=I$$ and take the determinant of both sides...their is a rule for taking the determinant of a product of matrices, and a rule for taking the determinant of the conjugate transpose of a matrix...use those rules!

 

[skrtic]

thanks for the help. i think i have it now.

i think i had it a while ago but didn't reason it to myself right.

i tried to prove a little more than i had to.