As the title says: How many matrices are defective?
My question is in analogy to the rationals and the irrationals, where there are strictly more irrationals (vastly, vastly more irrationals) than rationals. In fact, I think the analogy extends further in the sense that the rationals seem more numerous because almost all of our day-to-day interactions with numbers are interactions with rational numbers, even though the rationals are rare among the set of all reals.
Another analogy might be the singular and nonsingular matrices where, again, my understanding is that there is a strict sense in which there are more nonsingular matrices; in fact, it is my understanding that the word "singular" itself refers to the near-zero odds of selecting a singular matrix at random from the set of all matrices. And yet, here again, in our day-to-day lives, we often interact with nonsingular matrices, even though singular matrices are rare among the set of all matrices.
So, I am wondering if a similar situation occurs with the defective versus diagonalizable matrices? Put explicitly: Diagonalizable matrices seem rare to me and defective matrices seem abundant. But I am wondering if this is just an artifact of the way math is studied and discussed, and really the cardinality of the set of diagonalizable matrices is strictly larger than the cardinality of the set of defective matrices? Or no?
