# A mathematical derivation in Peskin and Schroeder on page 722.

• MathematicalPhysicist
In summary, the purpose of the mathematical derivation on page 722 in Peskin and Schroeder is to provide a rigorous explanation for a concept or equation in the field of physics, specifically within quantum field theory. The difficulty of understanding this derivation can vary from person to person, but with practice, it can be grasped by most individuals. The time it takes to complete a mathematical derivation can vary greatly and common mistakes include trying to memorize the steps without understanding the underlying concepts. To improve understanding, regular practice and seeking help from a mentor or tutor are recommended, along with a strong foundation in mathematics and physics fundamentals.
MathematicalPhysicist
Gold Member
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.

How do you quote nowadays in PF?

## 1. What is the purpose of a mathematical derivation?

A mathematical derivation is used to prove or derive a mathematical result or formula. It involves using logical steps and mathematical principles to arrive at a solution or conclusion.

## 2. Why is the mathematical derivation in Peskin and Schroeder on page 722 important?

The mathematical derivation on page 722 in Peskin and Schroeder's book is important because it provides a rigorous and systematic approach to understanding and solving problems in quantum field theory. It is a key tool for physicists and researchers in this field.

## 3. How do you approach a mathematical derivation?

To approach a mathematical derivation, you must first clearly define the problem or question you are trying to solve. Then, you can use known mathematical principles and techniques to manipulate equations and arrive at a solution. It is important to carefully justify each step and ensure that your reasoning is sound.

## 4. What are some common challenges in a mathematical derivation?

Some common challenges in a mathematical derivation include making assumptions or approximations that may not be valid, encountering complex equations or calculations, and reaching a dead end or incorrect solution. It is important to carefully check your work and seek help or guidance if needed.

## 5. How can I improve my skills in mathematical derivations?

To improve your skills in mathematical derivations, it is important to practice regularly and familiarize yourself with common techniques and principles. It can also be helpful to study and analyze derivations done by others, and to seek feedback and guidance from more experienced mathematicians or physicists.

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