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Eigenvalue and Eigenspace

  • Thread starter fk378
  • Start date
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1. Homework Statement
Let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace.

T is the transformation on R2 that reflects points across some line through the origin.



3. The Attempt at a Solution
Since they tell us that the point is reflected across the origin, I say that the eigenvalue= -1 and since T is a linear transformation, the eigenspace is in R2.

Are both the answer and reasoning correct?
 

Answers and Replies

Dick
Science Advisor
Homework Helper
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Vectors are differences between points, not the points themselves. In this case you should be thinking of the vector corresponding to (x,y) to be the arrow connecting (0,0) with (x,y). Now consider the two cases of vectors parallel to the line and perpendicular to the line separately.
 
367
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I don't think I understand how to find the eigenvalue when considering the parallel and perpendicular vectors.
 
Dick
Science Advisor
Homework Helper
26,258
618
Take an example, suppose the line is x=y. What are A((1,1)) and A((1,-1))? How did I pick those two vectors?
 

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