Chapter 3: Q6P (page 147)

Write out the proof of the Schwarz inequality (10.9) for a complex Euclidean space. Hint: Follow the proof of (10.4) in (10.5), replacing the definitions of norm and inner product in (10.1) and (10.2) by the definitions in (10.6) and (10.7). Remember that norms are real and $⩾ 0$ .

Short Answer

Expert verified

The Schwarz inequality for complex Euclidean space is

$| U . V | ⩽ \sqrt{{\sum_{i = 1}^{n} U_{i}}^{*} U_{i}} \cdot \sqrt{\sum_{i = 1}^{n} V_{i}^{*} V_{i}}$

Step by step solution

Given information from question

Schwarz inequality $= |\sum_{i = 1}^{n} A_{i}^{*} B_{i}| ⩽ \sqrt{\sum_{i = 1}^{n} A_{i}^{*} A_{i}} \sqrt{\sum_{i = 1}^{n} B_{i}^{*} B_{i}}$

Inner product and Norm

Inner product of $A$ and $B = \sum_{i = 1}^{n} {A_{i}}^{*} B_{i}$

Norm of $A = A = \sqrt{\sum_{i = 1}^{n} A_{i}^{*} A_{i}}$

Calculating the Schwarz inequality for complex Euclidean space

Consider the inner product and the norm for any given two vectors.

$U = (U_{1}, \dots \dots, U_{n}) V = (V_{1}, \dots . ., V_{n})$

The inner product $U . V = \sum_{}^{n} U_{i} * V_{i}$

The norm of $U$

$U = U = \sqrt{U_{1}^{2} + \dots \dots + U_{n}^{2}}$

The norm of $V$

$V = V = \sqrt{V_{1}^{2} + \dots \dots + V_{n}^{2}}$

Consider the dot below as a product between the same vector $(U + t V)$ . When it is taken within the same vector according to the definition of a dot product, it is always greater than zero.

$(U + t V) \cdot (U + t V) ⩾ 0 U \cdot U + t U \cdot V + t V \cdot U + t^{2} V \cdot V ⩾ 0 ||U^{2}|| + 2 t U \cdot V + t^{2} ||V^{2}|| ⩾ 0$

This takes the form of a quadratic equation $a t^{2} + b t + c ⩾ 0$ , where

$a = ||V^{2}||, b = 2 U . V$ , and $c = ||U^{2}||$ .

Solve for complex roots

Consider $Δ = b^{2} - 4 a c$ for complex roots. So,

$4 U^{2} V^{2} - 4 ||V^{2}|| ||U^{2}|| ⩽ 0 U V ⩽ \sqrt{||V^{2}|| ||U^{2}||} U . V ⩽ \sqrt{||V^{2}||} \cdot \sqrt{||U^{2}||}$

Thus, $| U . V | ⩽ \sqrt{{\sum_{i = 1}^{n} U_{i}}^{*} U_{i}} \cdot \sqrt{\sum_{i = 1}^{n} V_{i}^{*} V_{i}}$

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.

Write out the proof of the Schwarz inequality (10.9) for a complex Euclidean space. Hint: Follow the proof of (10.4) in (10.5), replacing the definitions of norm and inner product in (10.1) and (10.2) by the definitions in (10.6) and (10.7). Remember that norms are real and $⩾ 0$ .

Short Answer

Step by step solution

Given information from question

Inner product and Norm

Calculating the Schwarz inequality for complex Euclidean space

Solve for complex roots

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Torque and Rotational Motion

Mechanics and Materials

Energy Physics

Famous Physicists

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

Write out the proof of the Schwarz inequality (10.9) for a complex Euclidean space. Hint: Follow the proof of (10.4) in (10.5), replacing the definitions of norm and inner product in (10.1) and (10.2) by the definitions in (10.6) and (10.7). Remember that norms are real and ⩾0.

Short Answer

Step by step solution

Given information from question

Inner product and Norm

Calculating the Schwarz inequality for complex Euclidean space

Solve for complex roots

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Torque and Rotational Motion

Mechanics and Materials

Energy Physics

Famous Physicists

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

Write out the proof of the Schwarz inequality (10.9) for a complex Euclidean space. Hint: Follow the proof of (10.4) in (10.5), replacing the definitions of norm and inner product in (10.1) and (10.2) by the definitions in (10.6) and (10.7). Remember that norms are real and $⩾ 0$ .