Thread: Which of the following statements is true for the following matrix (eigenvalues)?

Let A be a matrix corresponding to projection in 2 dimensions onto the line generated by a vector v.

A. The vector v is an eigenvector for A corresponding to the eigenvalue lambda = −1.
B. Any vector w perpendicular to v is an eigenvector for A corresponding to the eigenvalue lambda = −1.
C. lambda = 1 is an eigenvalue for A
D. The vector v is an eigenvector for A corresponding to the eigenvalue lambda = 1.
E. lambda = 0 is an eigenvalue for A
F. None of the above

I know how to find eigenvalues and eigenvectors, but I don't really know what they correspond to. It's so abstract, and I'm unsure of how to see this question.

Can anyone point me in the right direction?

Thanks.

Originally Posted by swtdelicaterose

Let A be a matrix corresponding to projection in 2 dimensions onto the line generated by a vector v.

A. The vector v is an eigenvector for A corresponding to the eigenvalue lambda = −1.

Think "geometrically". Draw a picture. Draw a vector v and the line in its direction. What does "project onto that line" mean for a general vector in your picture? You are told that A projects any vector onto v. What is Av?

B. Any vector w perpendicular to v is an eigenvector for A corresponding to the eigenvalue lambda = −1.

If w is perpendicular to v, what is Aw?

C. lambda = 1 is an eigenvalue for A

A only has two eigenvalues and you found them in (A) and (B).

D. The vector v is an eigenvector for A corresponding to the eigenvalue lambda = 1.

Look at (A) again.

E. lambda = 0 is an eigenvalue for A

Look at (B) again.

F. None of the above

I know how to find eigenvalues and eigenvectors, but I don't really know what they correspond to. It's so abstract, and I'm unsure of how to see this question.

Can anyone point me in the right direction?

Thanks.