Idiosyncrasy: For the two tabs "Select Matrix A" and "Find Eigen" you may need to click at the very right-hand edge of the tab to select it.

Stage 1: "Select Matrix A": The domain is the left graph, and the image is shown at the right. If you have a particular matrix in mind, above the right-hand graph select the "arbitrary" transformation (if it isn't already selected) and drag the vectors T(u1) and T(u2) until the matrix columns are what you want. The unlabeled red vector is not needed, but you can try dragging it in the left-hand graph to see how it transforms.

Stage 2: "Find Eigen": In the left-hand graph (the domain), drag v around and see if you can arrange to have v and T(v) be along the same line. Then you have achieved T(v) = lambda v , an eigenvector with its eigenvalue. If you line them up really carefully, the eigenspace line will be shown.