 #1
MathematicalPhysicist
Gold Member
 4,699
 369
 Homework Statement:

Excuse my mathematical subtlety here.
But, they write the following:
[quote]
Define unitary matrices ##U_u## and ##W_u## by:
$$(20.135)\ \ \ \lambda_u \lambda_u^{\dagger}=U_u D_u^2 U_u^{\dagger} \ \ \ \lambda_u^{\dagger} \lambda_u = W_u D_u^2 W_u^{\dagger},$$
where ##D_u^2## is a diagonal matrix with positive eigenvalues.
Then:
$$(20.136) \ \ \ \lambda_u = U_u D_u W_u^{\dagger},$$
where ##D_u## is the diagonal matrix whose diagonal elements are the positive square roots of the eigenvalues of (20.135).
[/quote]
My problem is how to infer this direction, i.e that ##(20.135)\Rightarrow (20.136)##?, I can see how to infer the other direction, it's quite simple:
##\lambda_u = U_u D_u W_u^{\dagger} \Rightarrow \lambda_u\lambda_u^{\dagger}=U_u D_u W_u^{\dagger}W_u D_u U_u^{\dagger}=U_u D_u^2 U_u^{\dagger}##, and the same with the second identity in (20.135); but how do you get the other direction?
 Relevant Equations:
 The relevant equations are discussed in the problem statement.
My attempt at solution is in the HW template, though this is not an HW question.