Eigenvector of a matrix used as standard matrix of a linear transformation
First, set the definition of the matrix A by moving the two red vectors to wherever you like (or keep the default positions of the two red vectors). The transformation T from R^2 to R^2 is defined by the rule T(x)=Ax for any vector x in R^2.
Take the blue point, and move it. (You can move u where ever you want: in a variant version of this, u is restricted to have length 1.) The output of u, namely T(u) is shown in purple. If the vector u is chosen right, then you can see that u and T(v) point in the same (or in opposite) directions, which would make the vector u an eigenvector of A.