Finding Eigenvalues and Eigenspaces: A Reflection Transformation Example

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What does A scale them to?In summary, the given problem asked to find an eigenvalue of the matrix A without writing it, and to describe the eigenspace of A. The given information stated that A is the matrix of a linear transformation T, which reflects points across a line through the origin. The solution involves considering the vectors parallel and perpendicular to the line separately, and using the fact that T is a linear transformation. After considering an example, it was determined that the eigenvalue is -1 and the eigenspace is in R2.
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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.



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?
 
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  • #2
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.
 
  • #3
I don't think I understand how to find the eigenvalue when considering the parallel and perpendicular vectors.
 
  • #4
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?
 

1. What is an eigenvalue?

An eigenvalue is a scalar (single value) associated with a square matrix. It represents the scale factor by which a particular eigenvector is stretched or compressed when the matrix is applied to it.

2. How do you find the eigenvalues of a matrix?

To find the eigenvalues of a matrix, you first need to subtract the identity matrix multiplied by a scalar (λ) from the original matrix. Then, you set the determinant of this new matrix equal to 0 and solve for λ. The resulting values of λ are the eigenvalues of the original matrix.

3. What is an eigenspace?

An eigenspace is a vector space that contains all the eigenvectors associated with a particular eigenvalue. It is a subspace of the original vector space and can be thought of as the "direction" in which the matrix stretches or compresses the eigenvectors.

4. Why are eigenvalues and eigenspaces important?

Eigenvalues and eigenspaces are important because they help us understand the behavior of a matrix when applied to a vector. They can also be used to simplify complex matrices and make them easier to analyze and solve.

5. Can a matrix have multiple eigenvalues?

Yes, a matrix can have multiple eigenvalues. In fact, most matrices have multiple distinct eigenvalues. However, some matrices may have repeated eigenvalues, in which case the corresponding eigenvectors may be different.

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