Unitary Matrices: Finding the Most General Form

[fahd]

unitary matrices!please help!

HI

Iv got to find the most general form of a unitary matrix...

i.e taking a 2x2 matrix with components a,b,c,d., iv got to show a relationship between them so that all of the components can be related to one of the entry and a relationship is made which i can represent as a unitay matrix..
I managed to bring a relationship of two components with the other two...Can neone tell me is this it or more relations are possible?

thanks!

 

[Physics Monkey]

fahd,

A general 2x2 unitary matrix in has four free real parameters (so long as you don't restrict the determinant to be 1). A general 2x2 complex matrix has eight free real parameters (real and imaginary part of each entry). So you you need four real equations amongst your eight real parameters to be left with four free real parameters. So ask yourself, how many real equations do I have?

 

[fahd]

fahd,

A general 2x2 unitary matrix in has four free real parameters (so long as you don't restrict the determinant to be 1). A general 2x2 complex matrix has eight free real parameters (real and imaginary part of each entry). So you you need four real equations amongst your eight real parameters to be left with four free real parameters. So ask yourself, how many real equations do I have?

hi..
umm..actually iv bin asked to represent the general form of the unitary matrix assuming that all its entries are complex...sorry i forgot to mention this before!so i just have 4 equations in terms of the matrix's determinant!Now this is where I am stuck!please help!

 

[quantumdude]

Please show us what you've done. Then it will be easier for people to see where you're going wrong.

 

[Physics Monkey]

I am forbidden by the rules of the forum from saying anymore, but I can assure you that what I said applies to your case. Like Tom said, if you post your work maybe we can be more helpful.

 

[fahd]

I am forbidden by the rules of the forum from saying anymore, but I can assure you that what I said applies to your case. Like Tom said, if you post your work maybe we can be more helpful.

okie..heres what iv done.(attached image)...please help!

 

[Physics Monkey]

That looks good thus far, but you can get more out of your equations than you've done so far. What can you say about the determinant? Also, can a and b be as large as you like or are there restrictions imposed by your equations?

 

[fahd]

That looks good thus far, but you can get more out of your equations than you've done so far. What can you say about the determinant? Also, can a and b be as large as you like or are there restrictions imposed by your equations?

im seriously confused...everytime i try sumthing new...i get back to the result iv derived...wat do i do?

 

[Physics Monkey]

I don't know what to tell you except to keep plugging away. You can show, for instance, that the determinant has to be a complex number of absolute value 1. This is only a few lines of work from the equations you have written.

How comfortable are you with more advanced mathematics? If I was to start talking about matrix exponentials would you be able to follow?

 

[fahd]

I don't know what to tell you except to keep plugging away. You can show, for instance, that the determinant has to be a complex number of absolute value 1. This is only a few lines of work from the equations you have written.

How comfortable are you with more advanced mathematics? If I was to start talking about matrix exponentials would you be able to follow?

hi..
umm..besides the proof that iv shown u..i was also successful in provin'; that the determinant of the matrice should be equal to absolute value of 1 always.However I am just wondering that if i submit the answer that iv written down on tha attachment alongwith this condition of the determinant always being abs.1;will it be ok?I remember my prof saying that i might reach up a condition where i can find any 3 entries out of 4 on the basis of the 4th one..However in my case..im just able to get a relation between 2 entries and the other 2 entries..Is this sufficient...
Ah! as far as exponential matrices go...umm..i think i know that in case of unitary matrices..we can write the det as e^ic where c is a real number////but we haven't dun much of this..so I am not tooo good wid this!

do u think is shud go with the answer i got?And ya.the answer that i got..has 'w' -(determinant) included in it...I tried to sunstitute 'w' for the actual value..however that doesn't help maintain the relations that i have gotten between the entries!wat do i do?

 

[Physics Monkey]

The trouble with your answer so far is that since the determinant doesn't just depend on a and b, your unitary matrix has too many parameters. The determinant is equivalent to one real parameter since it has absolute value one. But this gives 2 (real and imaginary parts of a) + 2 (real and imaginary parts of b) + 1 (phase of determinant) = 5 which is too many parameters. The key is that there is some constraint on the absolute value of a and b that you have missed thus far. You are very close however.

With regards to the matrix exponential, what you can say is that any unitary matrix is the exponential of an anti-hermitian matrix. It is a lot easier to parameterize all the possible anti-hermitian matrices but this is equivalent to parameterizing all the unitary matrices (through the exponential function). It turns out there are four real parameters that describe all possible anti-hermitian matrices, the same number as descibe unitary matrices exactly as it must be.