The Eigen-things Assignment. Below you will find a list of 2-by-2 matrices M and vectors v. Each matrix has a parameter r, for which you will need to consider 5 different values.

1. Locate your own personal, complete set with matrix, vector, and pair of parameters.
2. For yourself, chose three additional values of the parameter: one value (r₁) smaller than r₂, one value (r₃) between r₂ and r₄, and one value (r₅) larger than r₄.
3. For each of these five values, say something about the matrix M:
1. What is the determinant of M?
2. What are the eigenvalue(s), eigenvector(s), and generalized eigenvector (if any) of M?
   
4. For each of these five values, you should analyze the affine transformation T(x) = Mx + v.
   
1. Are there fixed points? (A fixed point for an affine transformation is a point that gets sent to itself: T(x) = x.) 
2. What does T do to the plane or to the unit square?
   
3. Draw a picture of what T does to the plane using either a unit square or a parallelogram based on eigenvectors.
4. Similarly for each of these values, you should analyze the differential equation D(x) = Mx + v. 
1. Are there any equilibria points for the differential equation?
2. What is the general solution?
   
3. What does the phase plane look like?
   

	M	v	r2	r4
Brendan	6 r 2 -2	-60 -120	-8	-6
Claire	1 r -1 -5	-60 -120	5	9
Dan L.	6 r 1 2	-60 -120	-4	12
Dave Balderston	4 r 1 -2	-60 -120	-9	-8
Dave Birchler	3 r 1 -1	-60 -120	-4	-3
Frank	2 r 1 1	-60 -120	-0.25	2
Giabao	2 r -1 4	-60 -120	-8	1
G.W.	3 r -1 1	-60 -120	-3	1
Julie	3 r 1 7	-60 -120	-4	21
J.B.	5 r 1 3	60 -120	-1	15
Laura H.	-6 r 2 -2	60 120	-2	6
Laurie B.	-6 r 3 -6	60 120	0	12
Mariah	-6 r 1 2	60 -120	-16	-12
Matt	-4 r 1 -2	60 120	-1	8
Nathan	-3 r 1 -1	60 120	-1	3
Nick	-10 r 5 0	60 120	-5	0
Rudy	2 r 1 4	-60 -120	-1	8
Shawn	-3 r -1 1	60 -120	3	4
Steph	-3 r 1 1	60 -120	-4	-3
Ted	3 r -1 1	-120 -60	-3	1
Thomas	3 r 1 7	-60 -120	-4	21
Yuan Alex	-1 r 1 5	-120 60	-9	-5
Robinson	-6 r 2 -2	120 -60	-2	6
Romig	-6 r 3 -6	120 120	0	12

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