> 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.
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.
I've added a caveat along the lines of your suggestion.
> 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.
It wasn't meant to be complacently terse, it was meant to represent the basic tautology. Mathematics is a language as well as a way of thinking and one needs both since things are truly complicated under the hood. The language is actually our friend here, it allows people to speak clearly and have near universal understanding of what is meant. With this in mind, the correct answer to 'how do we communicate maths to non-mathematicians'(as opposed merely to teaching its importance which is advocacy) is to teach them the maths. That's how I see it anyway.
I understand the tautology. It's inherent in every field that requires specialization. And the point I'm trying to make is that the best teaching relies on ideas or analogies already familiar to the student. It's not enough to say math is a language. So is French. But to learn French, we first learn to map our native language to the new one. The analogy isn't perfect, but math often describes things in the world, and we can use those things to provide windows to the math itself.
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. ;)
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.