- #1

MathematicalPhysicist

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- 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.