unitary matrix

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

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

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

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

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

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


So start with [tex]RR^{*}=I[/tex] 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.