Question

Do Eigenvectors and values only have meaning for square matrices?

Answer 1

Yes, same as inverses.

Answer 2

No no no. They are useful for way more than that! :)

For a given linear transformation the eigenvectors represent a convenient basis for the range. Consider the image of a circle (or sphere). After the transformation, the circle is mapped to an ellipse (or 3d ellipsoid) The eigenvectors represent the major/minor axes and the values give the lengths of those axes. :)

Evalues and Evectors are SUPER useful in differential equations as they can help you find the fundamental solution set (which is the basis for the solution space). The eigenvalues serve a similar purpose -- they scale the eigenvectors appropriately. You can write the solution to any differential equation in terms of these fundamental solutions.

Answer 3

True^^ But even in diffeq you have to have a square matrix to find e-values (in turn, e-vectors). :P

Answer 4

Oh man. I read "square" as 2x2. :-p Correct you are malevolence.