Graphical Representation of the Eigenvalues and Eigenvectors

The eigenvectors of a tensor $M:{\mathrm{R}}^n\Rightarrow {\mathrm{R}}^{n}M:\; ℝ^n\; \Rightarrow \; ℝ^n$ are those vectors that do not change their direction upon transformation with the tensor M but their length is rather magnified or reduced by a factor λ. Notice that an eigenvalue can be negative (i.e., the transformed vector can have an opposite direction). Additionally, an eigenvalue can have the value of 0. In that case, the eigenvector is an element of the kernel of the tensor.

The following example illustrates this concept. Choose four entries for the matrix $M:{\mathrm{R}}^2⟹{\mathrm{R}}^{2}M:\; ℝ^2\; ⟹\; ℝ^2$ and press submit. The tool then draws 8 coloured vectors across the circle and their respective images across the ellipse. Use visual inspection to identify which vectors keep their original direction. The tool also finds at most two eigenvectors (if they exist) and draws them in black along with their opposite directions.