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Chapter 7: Q6E (page 323)

Question: If a vectorv鈫�is an eigenvector of both AandB, isv鈫� necessarily an eigenvector ofAB?

Short Answer

Expert verified

Yes, the given statement is True.

Step by step solution

01

Define the eigenvector

Eigenvector:An eigenvector of Ais a nonzero vector vinRnsuch thatAv=位惫, for some scalar位.

02

Given data

Consider v鈫�is an Eigen vector of A and B.

And the given statement is v鈫�necessarily an eigenvector of AB.

The objective is to write the given statement is True or False.

03

Step 3:Check whether v→ is necessarily an eigenvector of AB

If v鈫�is an Eigen vector of A and B this means that:

Av鈫�=伪v鈫�and

Whereandare eigenvalues ofandrespectively, then

(AB)v鈫�=A(Bv鈫�)尾v鈫�=尾(Av鈫�)Av鈫�=伪v鈫�=尾伪v鈫�=伪尾v鈫�sinceBv鈫�=尾v鈫�=尾Av鈫�sinceAv鈫�=伪v鈫�=尾伪v鈫�

Thus, the eigenvector for AB is v鈫�.

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Most popular questions from this chapter

Consider the matrixA=[2034] Show that 2 and 4 are eigenvalues ofand find all corresponding eigenvectors. Find an eigen basis for Aand thus diagonalizeA.

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Reflection about a line L inR2.
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