Eigenvalues and eigenvectors
Ah, yes – we have arrived at the meat of this chapter, the eigen-stuff! We know that linear transformations are involved in this somehow, but how? Well, let's start with something we are familiar with, the reflection transformation we covered in Chapter 5, Using Matrices to Transform Space. Here is the diagram that describes it:
Figure 7.2 – Reflection transformation
Now, the special vectors we are looking for, called eigenvectors, are the ones that only get multiplied by a scalar when they are transformed. Here's a look at a number of vectors being reflected, with the reflections being the dashed lines. Stare at the diagram and see if you see any reflections that are the same as multiplying by a scalar:
Figure 7.3 – Vectors and their reflections
Well, I don't want you to stare too long, so here is the first one:
Figure 7.4 – The first eigenvector...
