What are the differences with sparse and dense matrices in practice, so as to offer some insight to new learners on a more intuitive level.
Obviously everyone knows about the dictionary definition of sparse and dense matrices (a definition based on the portion of zero/non-zero elements)
But why are they so important from a mathematical application/optimization/problem solving point of view?
Is it that a lot of neat algorithms are defined such that they can only be operated on a problem if it satisfies such and such criteria, and some guy just proved that sparse | dense matrices tends to satisfy the aforementioned criteria really well
Or is it to do with the limited amount of computer memory available in real life, and that we must somehow "compress" matrices for faster computation - as such sparse matrices would be more desired
Or is it just a fuzzy guideline word that mathematicians use, as opposed to strict criterion fulfilling definitions that imply X properties about the matrices (e.g. make sure matrice is sparse and not dense because too many elements/variables too long to compute - or something to that nature?)
is the only major difference as a result of computational limitation and resource savings or are there fundamental mathematical differences between the two that make one uniquely operable and the other not
so essentially it revolves around our ability to compute something. so there really isnt some "fundamental" difference (like the difference between the first derivative or a second derivative of a function). but its just a thing that rose out of technical limitations in real life during computation.