Lecture 12 - Linear Maps of the plane and the Matrix Exponential

So for tomorrow, we will finish classifying the orbit structure of linear maps of the plane, basically concluding that orbits of planar linear maps follow similar patterns as that found in the trajectories of linear, homogeneous, autonomous ODEs in 2-dimensions.

But there is a twist, in that it is the sign of the eigenvalues that determines the type and stability of the origin, as an isolated equilibrium of a nondegenerate linear system.  But it is the magnitude of the eigenvalues of a linear map from the plane to itself, that determines the type and stability of the isolated fixed point at the origin. 

The secret as to why lies in calculating the time-1 map of a linear system of ODEs.  THAT is a linear map from the plane to the plane.  But that linear map does NOT have the same matrix as the linear system.  It is actually the matrix exponential of the matrix that determines the linear system, and that is sometimes a bit difficult to calculate.  We will spend a short time on this. 

If there is time, we may also move on to some thing new:  map on the circle.  See you tomorrow. 

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