For an invertible matrix A, prove that A and A have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A¹? Letting x be an eigenvector of A gives Ax = λx for a corresponding eigenvalue λ. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = λx Ax = λx Ax = λx Ax = λx A/(Ax) = A/(2x) Ax/A = 2x/A A-¹AX = A-¹2x AXA-¹ = 2XA-1 0044 O(A/A)X = (A/2)x O (A/A)X = 2x4-1 OxAA-1 = 24-¹x Ix = 2A-¹x Ix = (A/2)x Ix = 2xA-1 XI = 2A-¹x x = 2A-¹x x = 2A-¹x x = 2x4-1 x = 2A-¹x A-¹x = 1x A-¹x = 1x 2 A-¹x = 1x * 2 A-¹x = 1x λ This shows that --Select--- is an eigenvector of A-1 with eigenvalue ---Select---

For an invertible matrix A, prove that A and A have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A¹? Letting x be an eigenvector of A gives Ax = λx for a corresponding eigenvalue λ. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = λx Ax = λx Ax = λx Ax = λx A/(Ax) = A/(2x) Ax/A = 2x/A A-¹AX = A-¹2x AXA-¹ = 2XA-1 0044 O(A/A)X = (A/2)x O (A/A)X = 2x4-1 OxAA-1 = 24-¹x Ix = 2A-¹x Ix = (A/2)x Ix = 2xA-1 XI = 2A-¹x x = 2A-¹x x = 2A-¹x x = 2x4-1 x = 2A-¹x A-¹x = 1x A-¹x = 1x 2 A-¹x = 1x * 2 A-¹x = 1x λ This shows that --Select--- is an eigenvector of A-1 with eigenvalue ---Select---

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices

Problem 41EQ

See similar textbooks

Similar questions

Show that A=[0110] has no real eigenvalues.
Show that no 22 matrices A and B exist that satisfy the matrix equation. AB-BA=1001.
11. Find two nonzero matrices and such that.

Recommended textbooks for you

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning