Let A be the matrix of the linear transformation T R2R2 that

Let A be the matrix of the linear transformation T: R2->R2 that reflects each point across the x1-axis. What are the eigenvectors and corresponding eigenvalues of A? Hint: Begin by thinking about which points don\'t get moved by the linear transformation, and which points get moved to the negative of their original position. It is always helpful to observe the effect that T has a box with its lower left corner at the origin.

Solution

Let T be the linear transformation from R^2 to R^2 that reflects points about the X-axis.

Then T(x,y)=(x,-y)

The points on the X-axis are not affected by T as T(x,0)=(x,0)

Thus, points of the form (x,0) are eigenvectors of T with eigenvalue 1.

The points on the Y-axis are moved to the negative of their original co-ordinates as:

T(0,y)=(0,-y)=-(0,y).

Thus, points of the form (0,y) are also eigenvectors of T with eigenvalue -1.