1. ## Eigenvalue problems

Conder the matrix
A=[cos B -sin Q
sin B cos B ]

Multiplying a vector x in R^2 by A has the effect of rotationg x by angle B counter-clockwise about the origin.

1. Reasoning geometrically, give two values of B in [ 0,2pi) for which A has real eigenvalues. For each such B, state the eigenvalues and corresponding eigenvalues associated with them.

2.There is a value of B in [0,pi] for which A has i as an eigenvector. Find this value of B and find an eigenvector corresponding to the eigenvalue i in this case.

3. The product of the eigenvalues of a matrix are always equal to the determinant of the matrix. Use this fact to determine the other eigenvalue of A for the value of B u find in part 2. Find a corresponding eigenvector.

2. Originally Posted by shannon1111
Conder the matrix
A=[cos B -sin Q
sin B cos B ]

Multiplying a vector x in R^2 by A has the effect of rotationg x by angle B counter-clockwise about the origin.

1. Reasoning geometrically, give two values of B in [ 0,2pi) for which A has real eigenvalues. For each such B, state the eigenvalues and corresponding eigenvalues associated with them.
A rotation keeps lengths the same so if $\displaystyle Av= \lambda v$, there are only two possible values of $\displaystyle \lambda$. And then it should be clear what the two angles are. (If one vector is a multiple of the other they are parallel. v must be rotated into a parallel vector.)

2.There is a value of B in [0,pi] for which A has i as an eigenvector. Find this value of B and find an eigenvector corresponding to the eigenvalue i in this case.
To get i as an eigenvalue (NOT eigenvector) we must be thinking of (x,y) as representing the complex number x+iy. Av= iv becomes A(x+iy)= i(x+iy). What is i(x+iy)? What point does that correspond to? What angle is (x,y) rotated through?

3. The product of the eigenvalues of a matrix are always equal to the determinant of the matrix. Use this fact to determine the other eigenvalue of A for the value of B u find in part 2. Find a corresponding eigenvector.
All rotations have the same determinant. What is it? (Hint: the identity transformation:x-> x is a rotation through 0 degrees.)