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Linear Algebra With Applications
Found in: Page 324
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Show that similar matrices have the same eigenvalues. Hint:v If is an eigenvector of S-1AS, then role="math" localid="1659529994406" Sv is an eigenvector of A.

We have proved that similar matrices have the same eigenvalues.

See the step by step solution

Step by Step Solution

Step 1: 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.

Step 2: Find eigenvalue

Assume, that is an Eigen vector forS1AS .

Therefore, by definition:


Now manipulate the above equation as shown below:

S-1ASv^=λvrrSS-1ASv=rr(Multiplyby S fromLeft)ASv=λSrr(M)

From the above, the Eigen value of A is λand Svr is the Eigenvector.

Similarly, Eigen vector of A is Swu then S-1wu is an Eigen vector for S-1AS.

Hence, similar matrices have same eigen values.


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