This template demonstrates the effect of multiplying a vector by a matrix. Enter a 2 × 2 matrix and press Start. The vectors  and  are displayed and drawn. The vector  is the black vector in the lower graph, whereas the vector  is the black vector in the upper graph. The upper graph shows the orbit made by the tip of  as  traverses the unit circle. The eigenpairs for the matrix  are displayed on the left. If real, the normalized eigenvectors and their negatives are drawn in red and green in the lower graph. Moving the slider will change the angle the unit vector x makes with the positive -axis. As the slider is moved, the upper graph will display the vector  and the varying norm of  will be displayed on the left.

LinearAlgebra[Norm], LinearAlgebra[Eigenvectors], plots[arrow], plots[display]

Linear Algebra > Visualizations > EigenPlot Tutor

LinearAlgebra[Eigenvalues], LinearAlgebra[Eigenvectors]