Linear Algebra Question: Eigenvectors & Eignvalues

[FrogginTeach]

This is my last week in Linear Algebra. I am working on our last homework assignment before the exam so I want to make sure I know what I am doing.

In each part, make a conjecture about the eigenvectors and eigenvalues of the matrix A corresponding to the given transformation by considering the geometric properties of multiplication by A. Confirm each of your conjectures with computations.

A. Reflection about the line y=x
B. Contraction by a factor of 1/2

I'm not quite sure how to get started.

[HallsofIvy]

reflection about a line does not change the length of a vector. That sharply restricts the possible eigenvalues. Also consider two crucial cases:
(1) reflection about y= x of the vector <1, 1>, lying in y= x.
(2) reflection about y= x of the vector <1, -1>, perpendicular to y= x.

For the second, think about happens to <x, y>. "Contraction by a factor of 1/2" changes <x, y> to what vector? How can that be equal to \lambda <x, y>?