Why do we want lots of $0$’s in a matrix? I am working thru Axler’s “Linear Algebra Done Right.”  I am okay with the math, but I’m losing sight of the forest.  
Like all books on linear algebra, there’s a lot of time and energy spent on finding, proving existence of, and interpreting matrices with “lots of zeros” - that is, matrices with as many zeros as possible given some particular vector space and transformation. But I cannot see why a simple matrix generally, or one with lots of 0’s in particular, is very important.
Furthermore, discussions of matrices with lots of zeros closely correspond to discussions of the eigenvalues and eigenvectors (or generalized eigenvectors) of the transformation.  I see why eigenvectors and values are important for understanding a transformation, but we certainly don’t need a simple matrix to calculate the eigenvalues and vectors.
So, why are we spending so much time and energy finding matrices of the transformations with lots of zeros, and especially how such matrices relate to eigenvalues?
Given my lack of understanding as to why linear algebra procedes along this course, my question is necessarily vague.  Consequently, I am hoping only for a discussion of the issues, more so than specific mathematical derivations.
(Followup in response to comments):   And even if a sparse matrix allows easier computations, don’t the computations needed to find that sparse matrix generally negate any benefit?

There area a whole theory about SPARSE MATRICES  (matrices most of the elements are zero) which are used in computing science and numerical analysis. You can find many papers on this subject if you are interested in knowing more about it. 
