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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

Prove the part of Theorem 7.2.8 that concerns the trace: If an n 脳 n matrix A has n eigenvalues 位1, . . . , 位n, listed with their algebraic multiplicities, then tr A = 位1+路 路 路+位n.

Consider the matrix A=|abbc| where a, b, and c are nonzero constants. For which values of a, b, and c does A have two distinct eigenvalues?

Find a basis of the linear space Vof all2脳2matrices Afor which[1-3] is an eigenvector, and thus determine the dimension of V.

Consider a rotationT(x鈫�)=Ax鈫�in鈩�3in. (That is, A is an orthogonal 3x3matrix with determinant 1.) Show that T has a non-zero fixed point [i.e., a vectorT(x鈫�)=Ax鈫�in鈩�3withT(v鈫�)=v鈫�]. This result is known as Euler鈥檚 theorem, after the great Swiss mathematician Leonhard Euler (1707鈥�1783). Hint: Consider the characteristic polynomialrole="math" localid="1659595800447" fA(位). Pay attention to the intercepts with both axes. Use Theorem 7.1.4.

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.
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