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Chapter 7: Q48E (page 325)

IfAis a matrix of rank 1, show that any non-zero vector in the image of Ais an eigenvector of A.

Short Answer

Expert verified

Any non-zero vector in the image of A is an eigenvector of A.

Step by step solution

01

Definition of the image of a matrix

The image of a linear transformation or matrix is the span of the vectors of the linear transformation.

02

Checking whether any non-zero vector in the image of  is an eigenvector of

If is of rank 1, then dim mA = 1.

Therefore, for any vector v鈫�鈭�imA, the vectors v andAv鈫� must be colinear. Thus,v鈫� is an eigenvector of A.

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