anchor.diff:        :markdown
           Here, $1+i$ is an eigenvalue and $(1, i)$ is an eigenvector. 
   
  -        Whenever a matrix has complex eigenvalues, the system it describes will spiral around $(0,0)$. To see this, drag $A$'s columns (the arrows) around until you get a spiral. The eigenvalues are plotted in the real/imaginary plane to the right. You'll see that whenever the eigenvalues have an imaginary part, the system spirals, no matter where you start things off.
  +        If a matrix has complex eigenvalues, its sequence spirals around $(0,0)$. To see this, drag $A$'s columns (the arrows) around until you get a spiral. The eigenvalues are plotted in the real/imaginary plane to the right. You'll see that whenever the eigenvalues have an imaginary part, the system spirals, no matter where you start things off.
   
       section(ng-controller="FourQuadCtrl")
         | steps:
  
positive.code: 
        p
          :markdown
            It turns out that a matrix like $A$, whose rows add up to zero (try it!), is called a [Markov matrix](http://www.math.harvard.edu/~knill/teaching/math19b_2011/handouts/lecture33.pdf), and it always has $\lambda = 1$ as an eigenvalue. That means there's a value of $v_t$ for which $Av_t =\lambda v_t = 1 v_t = v_t$. At this "steady state," the number of New Yorkers moving to California exactly equals the number moving from California to New York, and the two populations stay the same forever. Hover over the animation to see the system go to the steady state.
  
        div(ng-controller="StochasticMatrixMultiplicationCtrl" style="position:relative")
          migration(style="width:70%;float:left")
  
negative.code: 
          :markdown
  
          p
            :markdown
              If you can draw a line through $(0,0)$, $v$ and $Av$, then $Av$ is just $v$ multiplied by number $\lambda$; that is, $Av = \lambda v$. In this case, we call $\lambda$ an __eigenvalue__ and $v$ an __eigenvector__. For example, here $(1,2)$ is an eigvector and $5$ an eigenvalue.
  