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Chapter 7: Q49E (page 384)

TRUE OR FALSE

If the rank of a square matrix A is 1, then all the nonzero vectors in the image of A are eigenvectors of A.

Short Answer

Expert verified

True, all the nonzero vectors in the image of A are eigenvectors of A.

Step by step solution

01

Define eigenvector

For any scaler 位 , if Av鈫� = 位v鈫� , then v鈫� is the eigenvector of matrix A.

02

Explanation for all the nonzero vectors in the image of A are eigenvectors of A:

Maybe because linearity would allow a zero vector to be an eigenvector and allow any eigenvalue.

If rankA = 1 then , 鈭� v鈭� Rn,im A = span (v).

Thus, x鈭� im A

Then, we have x = 伪v and Ax = A(伪v)= 尾v = (尾 /伪) 伪v

So, every vector in im A is an eigenvector of A.

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