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Matrix/2x2/Shearing matrices/Eigenvalues/Example

From Wikiversity

We consider -shearing matrices

with . The condition for some to be an eigenvalue means

This yields the equations

For , we get and hence also , that is, only can be an eigenvalue. In this case, the second equation is fulfilled, and the first equation becomes

For , we get and thus is the eigenspace for the eigenvalue , and is an eigenvector which spans this eigenspace. For , we have the identity matrix, and the eigenspace for the eigenvalue is the total plane.