Eigenvectors and eigenvalues explained visually
I just tried to figure out the simplest rigorous explanation of linear transformations. Here's one in terms of straight lines. Let's say we have a transformation of the 2D plane, i.e. a mapping from points to points. We will call that a "linear transformation" if these conditions are satisfied:<p>1) The point (0, 0) gets mapped to itself.<p>2) Straight lines get mapped to straight lines, though maybe pointing in a different direction.<p>3) Pairs of parallel straight lines get mapped to pairs of parallel straight lines.<p>Hence the name "linear transformation" :-) We can see that all straight lines going through (0, 0) get mapped to straight lines going through (0, 0). Let's consider just those straight lines going through (0, 0) that get mapped to themselves. There are four possibilities:<p>1) There are no such lines, e.g. if the transformation is a rotation.<p>2) There is one such line, e.g. if the transformation is a skew.<p>3) There are two such lines, e.g. if the transformation is a stretch along some axis.<p>4) There are more than two such lines. In this case, you can prove that in fact all straight lines going through (0, 0) are mapped to themselves, and the transformation is a scaling.<p>Now let's consider what happens within a single such line that gets mapped to itself. You can prove that within a single such line, the transformation becomes a scaling by some constant factor. (That factor could also be negative, which corresponds to flipping the direction of the line.) Let's call these factors the "eigenvalues", or "own values" of the transformation.<p>Now let's define the "eigenspaces", or "own spaces" of the transformation, corresponding to each eigenvalue. An eigenspace is the set of all points in the 2D plane for which the transformation becomes scaling by an eigenvalue. Let's see what happens in each of the cases:<p>1) In case 1, there are no eigenspaces and no eigenvalues.<p>2) In case 2, there is only one eigenspace, which is the straight line corresponding to the single eigenvalue.<p>3) In case 3, it pays off to be careful! First we need to check what happens if the two eigenvalues are equal. If that happens, it's easy to prove that we end up in case 4 instead. Otherwise there are two different eigenvalues, and their eigenspaces are two different straight lines.<p>4) In case 4, the eigenspace is the whole 2D plane.<p>In this way, eigenvalues and eigenspaces are unambiguously geometrically defined, and don't require coordinates or matrices.<p>Now, what are "eigenvectors", or "own vectors" of the transformation? Let's say that an "eigenvector" is any vector for which our transformation is a scaling. In other words, an "eigenvector" is a vector from (0, 0) to any point in an eigenspace. The disadvantage is that it involves an arbitrary choice. The advantage is that eigenvectors can be specified by coordinates, so you can find them by computational methods.<p>Does that make sense?