Q7.6-41E Find a 2 脳 2 matrix A without r... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q7.6-41E (page 382)

Find a 2 脳 2 matrix A without real eigenvalues and a vector ${\overset{鈫�}{x}}_{0} i n R^{2}$ such that for all positive integers t, the point $A^{t} {\overset{鈫�}{x}}_{0}$ is located on the ellipse in the accompanying sketch.

Short Answer

Expert verified

The 2 脳 2 matrix A without real eigenvalues $A = [\begin{matrix} \frac{18}{25} & \frac{- 73}{50} \\ \frac{26}{25} & \frac{- 18}{25} \end{matrix}]$ and a vector ${\overset{鈫�}{x}}_{0} i n R^{2}$ isrole="math" localid="1668431614671" $x_{0} = [\begin{matrix} 8 \\ 6 \end{matrix}]$

Step by step solution

Define eigenvalue:

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue, often denoted by $位$ .

Find 2 × 2 matrix A and vector:

From figure we have,

$v_{1} = [\begin{matrix} 8 \\ 6 \end{matrix}] v_{2} = [\begin{matrix} - 3 \\ 4 \end{matrix}]$

As is it perpendicular,

$A v_{1} = v_{2} A v_{2} = - v_{1} (\begin{matrix} a & b \\ c & d \end{matrix}) [\begin{matrix} 8 \\ 6 \end{matrix}] = [\begin{matrix} - 3 \\ 4 \end{matrix}] (\begin{matrix} a & b \\ c & d \end{matrix}) [\begin{matrix} - 3 \\ 4 \end{matrix}] = [\begin{matrix} - 8 \\ - 6 \end{matrix}]$

Multiply the LHS,

$8 a + 6 b = - 3 8 c + 6 d = 4 - 3 a + 4 b = - 8 - 3 c + 4 d = - 6$

Find 2 × 2 matrix A without real eigenvalues:

So, the 2 脳 2 matrix A without real eigenvalues $A = [\begin{matrix} \frac{18}{25} & \frac{- 73}{50} \\ \frac{26}{25} & \frac{- 18}{25} \end{matrix}]$

the eigenvalues of A are $位_{1, 2} = \frac{18 卤 \sqrt{949} i}{25}$

So, the vector is $x_{0} = [\begin{matrix} 8 \\ 6 \end{matrix}]$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find a 2 脳 2 matrix A without real eigenvalues and a vector ${\overset{鈫�}{x}}_{0} i n R^{2}$ such that for all positive integers t, the point $A^{t} {\overset{鈫�}{x}}_{0}$ is located on the ellipse in the accompanying sketch.

Short Answer

Step by step solution

Define eigenvalue:

Find 2 × 2 matrix A and vector:

Find 2 × 2 matrix A without real eigenvalues:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Calculus

Discrete Mathematics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Find a 2 脳 2 matrix A without real eigenvalues and a vector x鈫�0inR2such that for all positive integers t, the point Atx鈫�0is located on the ellipse in the accompanying sketch.

Short Answer

Step by step solution

Define eigenvalue:

Find 2 × 2 matrix A and vector:

Find 2 × 2 matrix A without real eigenvalues:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Calculus

Discrete Mathematics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

Find a 2 脳 2 matrix A without real eigenvalues and a vector ${\overset{鈫�}{x}}_{0} i n R^{2}$ such that for all positive integers t, the point $A^{t} {\overset{鈫�}{x}}_{0}$ is located on the ellipse in the accompanying sketch.