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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?

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    $\begingroup$ (1) we don't need a simple matrix to calculate eigenvalues, but, if the matrix is very large, zero entries will save computation time; (2) if a matrix has only zeroes below (or above) the diagonal, eigenvalues are on the diagonal $\endgroup$ – J. W. Tanner Aug 5 '19 at 22:11
  • $\begingroup$ I am not familiar with situations where computation time (whether to invert matrices, or find eigenvalues or vectors, or multiply matrices) is of any great importance. It could be helpful if you’d say more about situations where computation time would be an issue. $\endgroup$ – PossumP Aug 5 '19 at 22:27
  • $\begingroup$ I'm talking about real-world applications, such as matrices for large data sets, where it costs computer time to do each operation $\endgroup$ – J. W. Tanner Aug 5 '19 at 22:38
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    $\begingroup$ The pagerank algorithm Google uses, to oversimplify a bit, involves a matrix whose size is the number of (indexed) webpages on the internet. With huge real-world data sets like this that need to be manipulated mathematically, the sparseness is not insignificant. $\endgroup$ – runway44 Aug 5 '19 at 23:17
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    $\begingroup$ @Mpitts Computers are not nearly as fast as many would like, and there are a plethora of computations that are infeasible because of hardware limitations. One might even say that most computations are infeasible, and we're lucky to be able to perform one. $\endgroup$ – Matt Samuel Aug 6 '19 at 1:26
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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.

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Well, there are many matrices with lots of zeros, but most likely you're trying to refer to diagonal matrices. These have many important applications, and one other thing I recall is the fact that their determinants are easily calculated. Clearly one wants to see which type of matrices can be reduced to diagonal form, and so on.

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