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Math / Linear Algebra and Its Applications 5 / Chapter 5.8 / Problem 16E

Linear Algebra and Its Applications | 5th Edition | ISBN: 9780321982384 | Authors: David C. Lay; Steven R. Lay; Judi J. McDonald

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Suppose is an eigenvalue of the B in Exercise 15, and that

Chapter 5, Problem 16E

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QUESTION:

Problem 16E

Suppose is an eigenvalue of the B in Exercise 15, and that x is a corresponding eigenvector, so that Use this equation to find an eigenvalue of A in terms of and .

[Note: because B is invertible.]

Reference:

Suppose be a scalar different from the eigenvalues of A, and let from both sides of the equation , and use algebra to show that is an eigenvalue of B, with x a corresponding eigenvector.

Questions & Answers

QUESTION:

Problem 16E

Suppose is an eigenvalue of the B in Exercise 15, and that x is a corresponding eigenvector, so that Use this equation to find an eigenvalue of A in terms of and .

[Note: because B is invertible.]

Reference:

Suppose be a scalar different from the eigenvalues of A, and let from both sides of the equation , and use algebra to show that is an eigenvalue of B, with x a corresponding eigenvector.

ANSWER:

Solution 16E

Step 1

Consider that is an eigenvalue of the matrix corresponding to eigenvector.

Find an eigenvalue of in terms ofandas follows.

Then, by the definition of eigenvalue of , we have

,

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