# Proving Determinant of Unitary Matrix is Complex Number of Unit Modulus

• skrtic
In summary, the conversation discusses finding the determinant of a unitary matrix, which is a complex number with unit modulus. The expression for the modulus of a complex number is given, and the steps for finding the determinant of a unitary matrix are explained. The conversation also includes a student's attempt at solving the problem and receiving guidance.
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

Last edited:
skrtic said:
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.

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.

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

skrtic said:
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!

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.

## 1. What is a unitary matrix?

A unitary matrix is a square matrix with complex entries whose conjugate transpose is equal to its inverse. This means that multiplying a unitary matrix by its conjugate transpose will result in the identity matrix.

## 2. How do you determine the determinant of a unitary matrix?

The determinant of a unitary matrix can be calculated by taking the product of its eigenvalues. Since the eigenvalues of a unitary matrix have a modulus of 1, the determinant will also have a modulus of 1.

## 3. Why is the determinant of a unitary matrix always a complex number?

Since a unitary matrix has complex entries, its eigenvalues will also be complex numbers. The product of complex numbers is always a complex number, hence the determinant of a unitary matrix will always be a complex number.

## 4. What does it mean for the determinant of a unitary matrix to have a modulus of 1?

A complex number with a modulus of 1 lies on the unit circle in the complex plane. This means that the determinant of a unitary matrix has a magnitude of 1 and a phase angle of 0 or a multiple of 2π. In other words, it does not change the length of any vector that it operates on.

## 5. How is the concept of unit modulus related to the determinant of a unitary matrix?

The determinant of a unitary matrix having a modulus of 1 means that the matrix preserves the length of any vector it operates on. This is a fundamental property of unitary matrices and is important in various applications in physics and engineering.

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