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Matrices can be understood in different ways, e.g.

  1. Linear systems of equations
  2. (rich algebraic structure of) Linear mappings
  3. Graphs
  4. Evolution law of discrete-time Dynamical system

Well, 1. und 2. are the most prominent ones, canonically tought and widely understood - in particular, understanding each of these nurtures understanding the other one.

Similarly, 3. and 4. are fairly close. Consider as an example probability diffusion on a graph, which is usually modeled by iterated powers of a stochastic matrix.

It would be interesting to have correlate these two aspects of matrices. Of course, I am aware of algebraic graph theory, but I am not aware of a mutual enrichment of the "evolution perspective" (4.) with the "algebraic perspective". (1. & 2.). For example I would be interesting to have interpretations of the trace and the determinant from the former perspective.

Question: Is there theoretical research into that direction? Can show a good book about this topic?

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    $\begingroup$ just because they can both be described by matrices doesn't imply a meaningful relationship between the contexts as far as I can tell. $\endgroup$
    – Anthony Quas
    Jan 16, 2011 at 5:53
  • $\begingroup$ You might be interested in Dirichlet forms. $\endgroup$
    – Steve Huntsman
    Jan 16, 2011 at 6:46
  • $\begingroup$ Isn't spectral theory mainly a matter of dynamical properties of an operator? Starting e.g. with the formula for the spectral radius. $\endgroup$
    – Pietro Majer
    Jan 16, 2011 at 9:52
  • $\begingroup$ If you're interested in matrix differential equations, take a look at A List of Matrix Flows with Applications [pdf]. $\endgroup$
    – Rodrigo de Azevedo
    Dec 9, 2016 at 13:12

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I am pretty sure that this book talks about many of the things you are interested in:

@book {MR1369092,
    AUTHOR = {Lind, Douglas and Marcus, Brian},
     TITLE = {An introduction to symbolic dynamics and coding},  
 PUBLISHER = {Cambridge University Press},    
   ADDRESS = {Cambridge},
      YEAR = {1995},
     PAGES = {xvi+495},
      ISBN = {0-521-55124-2; 0-521-55900-6},    
   MRCLASS = {58F03 (15A48 54H20 58F20 94A24 94B60)},   
  MRNUMBER = {1369092 (97a:58050)}, MRREVIEWER = {Petr K{\.u}rka},
       DOI = {10.1017/CBO9780511626302},
       URL = { http://dx.doi.org/10.1017/CBO9780511626302 }, }
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  • $\begingroup$ @Willie: thanks for the formatting, did you do it by hand, or is there a trick? $\endgroup$
    – Igor Rivin
    Jan 16, 2011 at 23:58

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