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Chapter 7: Q35E (page 324)

Show that similar matrices have the same eigenvalues. Hint:v鈫� Ifis an eigenvector ofS-1AS, thenrole="math" localid="1659529994406" Sv鈫�is an eigenvector of A.

Short Answer

Expert verified

We have proved that similar matrices have the same eigenvalues.

Step by step solution

01

Definition of the Eigenvectors

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

02

Find eigenvalue

Assume, thatis an Eigen vector forS1AS.

Therefore, by definition:

S-1ASrv=位vr

Now manipulate the above equation as shown below:

S-1ASv^=位惫rrSS-1ASv=厂位rr鈥�鈥�鈥�(MultiplybySfromLeft)ASv=位厂rr鈥�鈥�鈥�(M)

From the above, the Eigen value of Ais 位and Svris the Eigenvector.

Similarly, Eigen vector of A is Swuthen S-1wuis an Eigen vector for S-1AS.

Hence, similar matrices have same eigen values.

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Most popular questions from this chapter

Consider a 4 脳 4 matrixA=|BC0D|where B, C, and D are 2 脳 2 matrices. What is the relationship among the eigenvalues of A, B, C, and D?

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if A is a 2脳2matrix with t r A = 5and det A = - 14what are the eigenvalues of A?

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For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

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