/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q6E Determine the definiteness of th... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q6E (page 400)

Determine the definiteness of the quadratic forms in Exercises 4 through 7.
6. $q (x_{1}, x_{2}) = 2 x_{1}^{2} + 6 x_{1} x_{2} + 4 x_{2}^{2}$

Short Answer

Expert verified

$位_{1} = 3 + \sqrt{10} and 位_{2} = 3 - \sqrt{10}$

The eigen values are negative and positive then the matrix is indefinite

Step by step solution

01

Given Information:

$q (x_{1}, x_{2}) = 2 x_{1}^{2} + 6 x_{1} x_{2} + 4 x_{2}^{2}$

02

Split the term and analyse the quadratic form:

$q (x_{1}, x_{2}) = 2 x_{1}^{2} + 6 x_{1} x_{2} + 4 x_{2}^{2}$

Split the term $6 x_{1} x_{2}$ equally between the two components are

$= [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] [\begin{matrix} 2 x_{1} + 3 x_{2} \\ 3 x_{1} + 4 x_{2} \end{matrix}]$

To analyse the quadratic form as

$q (\overset{鈫�}{x}) = \overset{鈫�}{x} 脳 A \overset{鈫�}{x}$

$A = [\begin{matrix} 2 & 3 \\ 3 & 4 \end{matrix}]$

role="math" localid="1659615148031" $\det (A - {位濒}_{2}) = 0 d e t ([\begin{matrix} 2 & 3 \\ 3 & 4 \end{matrix}] - 位 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]) = 0 d e t [\begin{matrix} 2 - 位 & 3 \\ 3 & 4 - 位 \end{matrix}] = 0 (2 - 位) (4 - 位) - (3) (3) = 0 8 - 2 位 - 4 位 + 位^{2} - 9 = 0 位^{2} - 6 位 - 1 = 0 (位 - (3 + \sqrt{10})) (位 - 3 - \sqrt{10)}) = 0$

$位_{1} = 3 + \sqrt{10} and 位_{2} = 3 - \sqrt{10}$

The eigen values are negative and positive then the matrix is indefinite

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider a linear transformation Tfrom ${鈩�}^{m} {鈩�}^{m}$ to ${鈩�}^{n} {鈩�}^{n}$ , where $m 鈮� n$ . Show that there exist an orthonormal basis ${\overset{鈫�}{v}}_{1}, . . ., {\overset{鈫�}{v}}_{m}$ of ${鈩�}^{m} {鈩�}^{m}$ and an orthonormal basis ${\overset{鈫�}{w}}_{1}, . . ., {\overset{鈫�}{w}}_{n}$ of ${鈩�}^{n}$ such that $T ({\overset{鈫�}{v}}_{i})$ is a scalar multiple of ${\overset{鈫�}{w}}_{i}$ , for i = 1,....m.

Hint: Exercise 19is helpful.

Find a symmetric 2x2matrix Bsuch that $B^{3} = \frac{1}{5} [\begin{matrix} 12 & 14 \\ 14 & 33 \end{matrix}]$

Show that for every indefinite quadratic form q on $R^{2}$ , there exists an orthogonal basis ${\overset{鈫�}{w}}_{1}, . . ., {\overset{鈫�}{w}}_{2}$ of such that $q (c_{1} {\overset{鈫�}{w}}_{1} + c_{2} {\overset{鈫�}{w}}_{2}) = c_{1}^{2} - c_{2}^{2}$ , Hint: Modify the approach outlined in

If Ais any symmetric 3x3matrix with eigenvalues -2,3, and 4, and $\overset{鈫�}{u}$ is a unit vector in ${鈩�}^{3}$ , what are the possible values of the dot product $\overset{鈫�}{u} . A \overset{鈫�}{u}$ ?

Matrix $[\begin{matrix} - 2 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix}]$ is negative definite.

See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.