Nasa JPL is using the framework for climatic modeling, so maybe we can help save the planet together. :)
One response I would have to both your comment and @alephone's rewording, which is perfectly correct, is that they are incomprehensible to most people. So the question is: How do you bridge the gap to make non-mathematicians understand?
The other would be: It's hard to talk about math. There's always slippage when you try to speak in an everyday way about technical things. I'm not sure how to get around that. Too much precision, in this case, undoes the goal of reaching another audience.
They are incomprehensible to those who do not understand the mathematics which are being discussed.
> So the question is: How do you bridge the gap to make non-mathematicians understand?
By teaching them mathematics.
> Too much precision, in this case, undoes the goal of reaching another audience.
No. In this case what failed was the author's understanding of the criterion for diagonalizability: the operator being normal or self-adjoint (depending on the underlying field). It isn't hard to add a caveat emptor saying "in some cases operators may not have a full set of eigenvectors, so it is only appropriate to say a matrix can have at most as many linearly independent eigenvectors as its order", provided the author realizes that this is indeed the case.
Precision is important because there are phenomenally important cases where operators have no eigenvectors at all! Consider an arbitrary rotation in R^2 (fun fact: in odd dimensions all real operators have an eigenvalue, why?); clearly, this fixes no vector.
> By teaching them mathematics.
The terse complacence of this statement, and your easy use of technical terms, make me doubt whether you know what it means to teach mathematics, to empathize with those who lack knowledge, and ultimately to help math exit the ghetto of jargon.
While the language of math is a useful tool for the initiated, like all linguistic barriers it keeps out newcomers. So yes, we must teach people math, but we must do it at first, as much as possible, in a language that allows them to enter into a new way of thinking. It's about unfurling a series of ideas one at a time instead of all at once. (I don't pretend to do that perfectly, but that was the intent. My understanding is as defective as some matrices. ;)
You should consider what happens with rotation matrices (no real eigenvectors), defective matrices (no complete set of eigenvectors), and the Jordan form. Wikipedia has adequate articles on these topics.
Also, about the Jordan form, if you've only ever worked with matrices over floats instead of exact precision, it would be understandable why you are not familiar with the Jordan form, as it's numerically useless, because it's numerically unstable. The tiniest perturbation will change a defective matrix into a diagonalisable matrix.
Or do you have a definition of independence for dimensions? That would first require a different definition of dimension. Ordinarily, dimension is just the cardinality of an independent, spanning set (we call this a basis). It is a single number. A linear space does not have a plurality of dimensions, nor do matrices have an associated plurality of dimensions.
Please don't mix technical words without defining them. In other areas of life, humans get along fine by using very imprecise, poetic, ambiguous, and metaphoric language. This is not how mathematics works. Everything you say must have a precise definition. :-(