I have figured out both of the confusing graphs. One was exactly what it should have been, so I’m a little disappointed that I ever stumbled over it. Truth to tell, I suppose I didn’t quite trust the authors, and that makes it harder to figure things out. Put it the other way: if I trust the author, I am sure that what I’m looking at is somehow right, and that limits the possibilities, making it far easier to reach an understanding.
The other graph was way more interesting. It is not consistent with the first graph. What makes it interesting is that that’s okay!
I can’t very well leave it at that, so ok, here’s a sketch – oops, I mean a summary, not a picture. (I do intend to discuss this example down the road.) The first graph requires that we have defined new variables; more precisely, it requires that we have defined a new basis for Rn. Given a new basis, I generally expect to compute new components for any given vector, i.e. components wrt (“with respect to”) the new basis. And there are simple matrix equations for the new basis and for the new components. (The word “simple” often means that I can write the equations from memory. But these equations are almost trivial. I will be discussing the change-of-basis equations, as they are called.)
But that’s not what they did. They rescaled the new components. Their equation for the components is not consistent with their equation for the new basis. But all they did was to change the scale on the axes. That’s pretty harmless.
I was all set to argue in general that scaling the eigenvectors (for that’s where they got their new basis) was relatively insignificant precisely because it only amounts to changing the scales on the axes. Ultimately, if the data analysis has merit, one would think that the merit doesn’t depend on the hash marks on the axes. I came down on the side of leaving the new basis vectors of unit length, but here I find that someone has chosen a different scale.
Having decided that the scale is relatively insignificant, I can have no complaint about their choosing a different one.
I could complain that their two graphs ought to be consistent, but I’ll let it go. Well, actually I’ll shout one thing from the rooftops: looking at someone else’s work, don’t bet the farm on these kinds of graphs being consistent! I don’t mind, but we need to know it. And they may not say it!
We all, of course, would choose to keep them consistent in our own work.
Anyway, with that issue out of the way, those graphs understood, I was able to collect all the results from all my sources. I need to get a lot of material ready for publishing here; and I need to dust off all the examples and make them presentable.
You know, I did find it hard to move on before I understood those graphs, even though it was other books I wanted to move on in. Odd. It would make sense if I couldn’t move on in that particular book, but why not move forward in a different book? Damfino.
All of that happened yesterday. Today was spent writing.