If I understand correctly, then any matrix can be diagonalized with real and +ve diagonal entries via a bi-unitary transformation. My question is, given a matrix, are the unitary matrices unique?? Is there any common textbook that provides a simple proof of this fact?
The next part of my question involves the quark sector of the Standard Model. In the gauge basis, the "mass matrix" of the quarks is a general complex one. We can then rotate the left and right handed fields separately to go to the mass basis. Now, my problem is that they are called "mass eigenstates" and the masses are called "eigenvalues". But, the Yukawa matrix in the original gauge basis is an arbitrary complex one which does not necessarily have "eigenvalues" in the usual sense of the term. So, is there any way to understand the meaning of the term "mass eigenstate" in this context ?
You can diagonalize M in general by first diagonalizing
M^dagger M and then diagonalizing M M^dagger. These
matrices are hermitian. Let us say that the first one is
diagonalised by a unitary transformation A
A (M^dagger M) A^-1 = diagonal
then you diagonalise M M^dagger by an unitary
transformation B
B (M M^dagger) B^-1 = diagonal
Then the biunitary transformation on M is
A M B^-1 = diagonal -------------> Eq.1
for quark mass matrices,
U_L M U_R^\dagger --> U_L=A and U_R=B
The proof is through making M upper/lower
triangular. You can see numerical recipes
algorithm part and also, many references are there.
You can diagonalize M in general by first diagonalizing
M^dagger M and then diagonalizing M M^dagger. These
matrices are hermitian. Let us say that the first one is
diagonalised by a unitary transformation A
A (M^dagger M) A^-1 = diagonal
then you diagonalise M M^dagger by an unitary
transformation B
B (M M^dagger) B^-1 = diagonal
Then the biunitary transformation on M is
A M B^-1 = diagonal -------------> Eq.1
for quark mass matrices,
U_L M U_R^\dagger --> U_L=A and U_R=B
The proof is through making M upper/lower
triangular. You can see numerical recipes
algorithm part and also, many references are there.
You can see a full proof on p357 of the book "Gauge Theory of Elementary Particle Physics" by Cheng and Li.
As stated above, the two unitary matrices which diagonalise M are those which diagonalize the two hermitian matrices MM* and M*M. And by the way, this is how you would go about numerically determining these matrices once you have fixed M.
Furthermore there is of course some phase freedom, so that you can left multiply both of the unitary matrices by the same diagonal phase matrix, let's call it P. Such that
P U_L M U_R* P* = P D P* = D
where D is the diagonal matrix with real and positive masses along the diagonal. Notice that left multiplying by P doesn't change the diagonalization at all. This is true for both the up and down quark mass matrices, so we have a total of 6 phase degrees of freedom in the quark sector. This phase freedom is used to remove the 5 non-physical phases from the CKM matrix.
Jonathan:
I was following your post. These issues are a bit
complicated for me. Let me see if I understand your
point.
To make eigenvalues real both U_L and U_R have to be
left multiplied by the same phase matrix P from left.
In CKM matrix, only P U_L will enter from the up sector.
Another Q D_L will enter from down sector and finally
P U_L (Q D_L)^dagger will give CKM matrix.
Hi Biswajoy,
I should have been more clear about that point, but you're exactly right.
Biswajoy Sir,
I think in your equation " A M B^-1 = diagonal & U_L M U_R^\dagger --> U_L=A and U_R=B", it should be like B M A^-1 = diagonal & U_L M U_R^\dagger --> U_L=B and U_R=A.Then it will satisfy previous equations.
B M A^\dagger=D.......................................(1)
A M^\dagger B^\dagger =D.......................(2)
B M A^\dagger A M^\dagger B^\dagger = D^2.............(3)
B M M^\dagger B^\dagger = D^2...........................(4)
Similarly,
AM^\dagger M A^\dagger = D^2...........................(4)
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