eigenvector
Most vectors get knocked off their original line when a matrix acts on them: the matrix both stretches and rotates them. An eigenvector is the rare exception. It is a special nonzero vector whose direction is left untouched. After the matrix acts, A*v still points along the same line as v; it may grow, shrink, or flip end-to-end, but it never swings off course.
Think of it as one of the natural axes of a transformation. If a sheet of rubber is stretched, the directions that simply lengthen or shorten without tilting are the eigenvectors. Find those special directions and you understand the heart of what the matrix does.
Example: for A = [[2,0],[0,3]], the vector (1,0) is an eigenvector because A*(1,0) = (2,0), which still lies on the x-axis. Same line, just twice as long.
(1,0) keeps its direction; the matrix only doubles its length.
An eigenvector must be nonzero, and any nonzero scalar multiple of an eigenvector is again an eigenvector for the same eigenvalue.