Explanation Given: The matrix A = [ − 6 1 3 − 3 0 2 − 20 2 10 ] and the vector x 0 = [ 1 1 − 1 ] . Approach: The expression for x k is given as, x k = A k x 0 = A k ( a 1 u 1 + a 2 u 2 + .... + a n u n ) = a 1 A k u 1 + a 2 A k u 2 + ........ + a n A k u n = a 1 ( λ 1 ) k u 1 + a 2 ( λ 2 ) k u 2 + ........ + a n ( λ n ) k u n Here, λ 1 , λ 2 , ........ , λ n are the eigenvalues and u 1 , u 2 , ...... , u n are the eigenvectors of the matrix A . Steps in Graphing Calculator: 1) Enter 2 nd + MATRIX key. 2) Press EDIT to enter the two matrices [ A ] and [ B ] . 3) Type the required equation needed to be calculated by pressing 2 nd + MATRIX key. 4) Press ENTER . Calculation: The sequence { x k } is generated by the following formula. x k = A x k − 1 Consider the matrix A = [ − 6 1 3 − 3 0 2 − 20 2 10 ] and the vector x 0 = [ 1 1 − 1 ] . Calculate the characteristic polynomial for A using the equation P ( t ) = | A − λ I | . Take | A − λ I | = 0 . | A − λ I | = 0 | [ − 6 1 3 − 3 0 2 − 20 2 10 ] − λ [ 1 0 0 0 1 0 0 0 1 ] | = 0 | [ − 6 1 3 − 3 0 2 − 20 2 10 ] − [ λ 0 0 0 λ 0 0 0 λ ] | = 0 | − 6 − λ 1 3 − 3 − λ 2 − 20 2 10 − λ | = 0 Solve further. λ 3 − 4 λ 2 − λ + 4 = 0 ( λ − 4 ) ( λ 2 − 1 ) = 0 ( λ − 4 ) ( λ − 1 ) ( λ + 1 ) = 0 λ = − 1 , 1 , 4 So, eigenvalues of A are λ 1 = − 1 , λ 2 = 1 and λ 3 = 4 . Find the corresponding eigenvectors. Consider, A x = λ 1 x ( [ − 6 1 3 − 3 0 2 − 20 2 10 ] − ( − 1 ) [ 1 0 0 0 1 0 0 0 1 ] ) x = 0 [ − 5 1 3 − 3 1 2 − 20 2 11 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] From above it can be concluded that, − 5 x 1 + x 2 + 3 x 3 = 0... ( 2 ) − 3 x 1 + x 2 + 2 x 3 = 0... ( 3 ) − 20 x 1 + 2 x 2 + 11 x 3 = 0... ( 4 ) From equation ( 2 ) . x 2 = 5 x 1 − x 3 Substitute the value of x 2 in equation ( 3 ) . 2 x 1 − x 3 = 0 x 3 = 2 x 1 Substitute the value of x 3 in equation ( 4 ) . x 1 = − x 2 That is, x 1 = − x 2 . First eigenvector is u 1 = [ 1 − 1 2 ] . Similarly, consider A x = λ 2 x to get the second Eigen vector u 2 = [ 1 1 2 ] . Also consider A x = λ 3 x , to get the third Eigen vector u 3 = [ 14 11 43 ] . Now, the starting vector x 0 can be expressed in terms of the eigenvectors as, x 0 = a 1 u 1 + a 2 u 2 + a 3 u 3 [ 1 1 − 1 ] = a 1 [ 1 − 1 2 ] + a 2 [ 1 1 2 ] + a 3 [ 14 11 43 ] From above equation it can be concluded that, a 1 + a 2 + 14 a 3 = 1

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

5th Edition

ISBN: 9780134689531

Author: Lee Johnson, Dean Riess, Jimmy Arnold

Publisher: PEARSON

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 4.8, Problem 14E

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An expression for xk by using the given equation, compute x4 and x10 with a calculator from the expression and comment on limk→∞xk.

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Chapter 4.8, Problem 13E

Chapter 4.8, Problem 15E

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Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

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Limits and Continuity; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=9brk313DjV8;License: Standard YouTube License, CC-BY

An expression for x k by using the given equation, compute x 4 and x 10 with a calculator from the expression and comment on lim k → ∞ x k .

Solution Summary: The author explains how to find an expression for x_k by using the given equation.

Concept explainers

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To find:

An expression for xk by using the given equation, compute x4 and x10 with a calculator from the expression and comment on limk→∞xk.

Want to see the full answer?

Chapter 4 Solutions

An expression for x k by using the given equation, compute x 4 and x 10 with a calculator from the expression and comment on lim k → ∞ x k .

Solution Summary: The author explains how to find an expression for x_k by using the given equation.

Concept explainers

Videos

To find: An expression for xk by using the given equation, compute x4 and x10 with a calculator from the expression and comment on limk→∞xk.

Want to see the full answer?

Chapter 4 Solutions

To find:

An expression for xk by using the given equation, compute x4 and x10 with a calculator from the expression and comment on limk→∞xk.