IMO eigenvectors are easiest to understand in connection with differential equations, and that's also one of their most important applications. If you plot the

IMO eigenvectors are easiest to understand in connection with differential equations, and that's also one of their most important applications. If you plot the flow of the equation x' = Ax then the eigenvectors are visually apparent. https://www.wolframalpha.com/input/?i=stream+plot+%7B-5x+%2B+3y,+3x+%2B+8y%7D https://www.wolframalpha.com/input/?i=stream+plot+%7B-5x+%2B... The eigendirections are the directions where the solution moves in a straight line. Not all matrices have (real valued) eigenvectors: https://www.wolframalpha.com/input/?i=stream+plot+%7Bx+%2B+5y,+-x+%2B+y%7D https://www.wolframalpha.com/input/?i=stream+plot+%7Bx+%2B+5...