Chapter 3: Q38P (page 160)

Verify equation (11.25). Hint: Remember from Section 9 that the transpose conjugate (dagger) of a product of matrices is the product of the transpose conjugates in reverse order and that $U^{+} = U - 1$ . Also remember that we have assumed real eigenvalues, so D is a real diagonal matrix.

Short Answer

Expert verified

$U^{- 1} M^{+} U = U^{- 1} M U \Rightarrow M^{+} = M$

Step by step solution

Given Information

$U^{- 1} M U = D$

Hermitian Matrix

A Hermitian matrix is a square matrix whose conjugate transpose matrix is equal to it. A Hermitian matrix's non-diagonal entries are all complex integers.

Hermitian Conjugate

Take the Hermitian conjugate of the first equation.

${(U^{- 1} M U)}^{+} = U^{+} M^{+} {(U^{+})}^{- 1} = U^{- 1} M^{+} {(U^{- 1})}^{- 1} [U^{+} = U^{- 1}] = U^{- 1} M^{+} U$

The eigenvalue of M is real and D is a real diagonal matrix.

$D^{+} = D$

Therefore,

$= U^{- 1} M^{+} U = U^{- 1} M U \Rightarrow M^{+} = M^{}$

Hence, verified.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Verify equation (11.25). Hint: Remember from Section 9 that the transpose conjugate (dagger) of a product of matrices is the product of the transpose conjugates in reverse order and that $U^{+} = U - 1$ . Also remember that we have assumed real eigenvalues, so D is a real diagonal matrix.

Short Answer

Step by step solution

Given Information

Hermitian Matrix

Hermitian Conjugate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Dynamics

Fundamentals of Physics

Modern Physics

Energy Physics

Astrophysics

Study anywhere. Anytime. Across all devices.

Verify equation (11.25). Hint: Remember from Section 9 that the transpose conjugate (dagger) of a product of matrices is the product of the transpose conjugates in reverse order and that U+=U-1. Also remember that we have assumed real eigenvalues, so D is a real diagonal matrix.

Short Answer

Step by step solution

Given Information

Hermitian Matrix

Hermitian Conjugate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Dynamics

Fundamentals of Physics

Modern Physics

Energy Physics

Astrophysics

Study anywhere. Anytime. Across all devices.

Verify equation (11.25). Hint: Remember from Section 9 that the transpose conjugate (dagger) of a product of matrices is the product of the transpose conjugates in reverse order and that $U^{+} = U - 1$ . Also remember that we have assumed real eigenvalues, so D is a real diagonal matrix.