Check my Understanding of Unitary and Diagonal Matrices

[RJLiberator]

Homework Statement

Find all diagonal unitary matrices.

Homework Equations

The Attempt at a Solution

I think I am starting to get the hang of this type of material.
I hope I am right in my thinking.

So if we have a diagonal matrix, let's say a 2x2 for a simple example:
<br /> \begin{pmatrix}<br /> a &amp; 0\\<br /> 0 &amp; b\\<br /> \end{pmatrix}

And we also have the condition that it is unitary, so the absolute values of the components must be equal to 1.

The general rule will then be:<br /> \begin{pmatrix}<br /> |a|=1 &amp; 0\\<br /> 0 &amp; |b|=1\\<br /> \end{pmatrix}

Where a,b ∈ℂ

And to make this more general, the pattern continues on for nxn matrices.

 

[andrewkirk]

Yes, but I imagine your lecturer expects you to express the matrices in parameterised form, rather than writing it as criteria as you have done above. You can give the general form of an n-dimensional, diagonal, unitary matrix using n parameters $\theta_1,\theta_2,...,\theta_n$. How would you write it?

 

[RJLiberator]

Ohhh, that is interesting. I believe I did see something like this when searching on the internet.

Would I write it out as e^(i*theta) multiplying the matrix :

<br /> e^{i\theta}<br /> \begin{pmatrix}<br /> a &amp; 0\\<br /> 0 &amp; b\\<br /> \end{pmatrix} where |a|^2=1 and |b|^2=1

How does that look?

 

[andrewkirk]

Getting closer. The trouble is that now you have three parameters (four, if we include the size of the matrix) and two criteria. What I was suggesting was writing it with two parameters (three if we include the size of the matrix) and no criteria, so that you can specify the set of all unitary, diagonal, n x n matrices as:

$$S_n=\{A\equiv (a_{ij})\in M(\mathbb{C},n)\ |\ A=diag\big(f_1(\theta_1),f_2(\theta_2),...,f_n(\theta_n) \big)\wedge \forall k:\theta_k\in\mathbb{C}\}$$

where $M(\mathbb{C},n)$ is the set of all n x n matrices of complex numbers and $diag(d_1,d_2,...,d_n)$ is the diagonal matrix with entries $d_1,d_2,...,d_n$ on the diagonal.

What are simple functions $f_1,...,f_n$ (not necessarily all different) that ensure that all the diagonal elements have modulus 1?

 

[RJLiberator]

I must admit that I feel lost looking at your notation, as if there is some barrier here.

Let me start with analyzing your question:
What are simple functions f1,...,fn (not necessarily all different) that ensure that all the diagonal elements have modulus 1?

Here, I believe this is beyond what my class is trying to get at.

Is there anything more 'simple' that can be looked at here?

 

[andrewkirk]

What are simple functions f1,...,fn (not necessarily all different) that ensure that all the diagonal elements have modulus 1?

Can you write a complex-valued function of $\theta$ that is guaranteed to have modulus 1 if $\theta$ is real? Can you get any complex number of modulus 1 as the value of such a function - ie by plugging in a suitable value of $\theta$?

 

[RJLiberator]

Wouldn't it be the values on the unit circle?

I am starting to understand what you are saying by modulus 1, but I am still looking at this wrong then.

Let me define something
Modulus: |λ| = sqrt(a^2+b^2)

So if modulus = 1, then a^2+b^2 = 1.
a being the real part, b being the imaginary part.

e^(i*theta) = cos(x)+isin(x)
So any angle will grab the value 1 as long as the radius is not greater than 1.

No?

 

[andrewkirk]

Yes, and since $re^{i\theta}$ is the complex number with modulus $r$ and argument (angle to positive real axis) $\theta$, the radius/modulus of $e^{i\theta}$ is...?

Then using that, can you write a diagonal 2 x 2 matrix in a form that guarantees that each of the two diagonal elements will have modulus 1?

 

[RJLiberator]

Interesting I like where this is going.

Modulus r = 1. So we have e^(i*theta)

The radius/modulus of e^(i*theta) = 1.

a 2x2 matrix in form that gurantees that each of the two diagonal elements will have modulus 1 is:

\begin{pmatrix}
e^(i*theta) & 0\\
0 & e^(i*theta)\\
\end{pmatrix}

Which we have the modulus = 1.

 

[andrewkirk]

It's not completely general yet. You can use different thetas for the 1,1 and the 2,2 elements. They don't have to be the same.

 

[RJLiberator]

Ohhh, that is making sense now, and I see why your notation above was as general as it was. As we can let theta have different values, but the form is e^(i*theta).