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A mathematical derivation in Peskin and Schroeder on page 722.
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[QUOTE="MathematicalPhysicist, post: 6311284, member: 72"] [B]Homework Statement:[/B] 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? [B]Relevant Equations:[/B] 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. [/QUOTE]
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A mathematical derivation in Peskin and Schroeder on page 722.
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