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Matrices and Graph

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Re: Matrices and Graph

#41

Post by qsdf38100 »

enriquto wrote: post #18
Scene_Cast2 wrote: post #15

Another interesting mapping is that a vector is (or can be thought of as) a discrete function (f(x) = ....) over an interval, a dot product of two vectors is a discrete integral product, and a matrix is a discrete scalar field. I wonder what the continuous form of a graph is... Some sort of a manifold perhaps?

> I wonder what the continuous form of a graph is... Some sort of a manifold perhaps? Exactly! The correspondence between manifolds and graphs is very beautiful. What many folks call today "graph signal processing" has traditionally been called "discrete differential geometry". Scalar fields are functions defined on vertices, vector fields are functions defined on edges, the incidence matrix is the gradient operator,…

And electromagnetism seems to be a by product of discrete differential geometry. Such a fascinating subject. Makes continuous treatment look like a mess.

Re: Matrices and Graph

#42

Post by algas »

dpflan wrote: post #6
dustingetz wrote: post #5

what book to read to develop these concepts and intuitions with engineering level of math (not proofs)?

I don't know how proof heavy this course is and it is not specialized on graphs: "Computational Science and Engineering I" -- Gilbert Strang (MIT) https://ocw.mit.edu/courses/18-085-computational-science-and...

The textbook for this course is probably one of the finest Strang ever wrote. It's also worth looking at the original edition of the textbook, which was called "Introduction to Applied Mathematics" and has a much more thorough treatment of the parallels between matrices, graphs, and differential operators, and their use in optimization problems. CSE is much more practical for solving actual scientific computing problems, though, even if I find the layout somewhat less beautiful.

Re: Matrices and Graph

#43

Post by FooBarBizBazz »

EsportToys wrote: post #9

Fun fact: this is only valid for domains that have a notion of "selfness", i.e. that there is such thing as an "identity matrix" for the quantities. Consider the following square matrix: TSLA APPL GOOG MSFT Alice | 100 5 0 1 Bob | 0 30 100 5 Carol | 2 2 2 2 Dan | 0 0 0 1000 An input vector of stock prices gives an output vector of net worths. However, that is about the only way you can use this matrix. You cannot tra…

I dunno, there are some semi-useful things you can do.

For example, the transform from (Alive, Bob, Carol, Dan) to (Male, Female) is linear -- it's another matrix that you can compose with the individual-ownership one you have here.

Or, call your individual-ownership matrix A, and say that P is the covariance of daily changes to prices of the four stocks listed. Then A P A' is the covariance of daily changes to the peoples' wealths. The framing as linear algebra hasn't been useless.

I kinda get what you're saying though. Like, why would powers of this matrix be useful? It only makes sense if there's some implicit transform between prices and people, or vice versa, that happens to be an identity matrix.

You can make up a story. Say the people can borrow on margin some fraction of their wealth. Then say that they use that borrowing to buy stock, and that that borrowing affects prices. Composing all these transforms, you could get from price to price, and then ask what the dynamics are as the function is iterated.

But, ok, "I'm just going to do an SVD of the matrix and put it in a slide" isn't going to tell anybody much.

Maybe there's a use for a rank-one approximation to this system? Like, "this is pretty close to a situation where there's a single ETF with those stocks in these proportions, and where the people own the following numbers of shares in the ETF"? Maybe if you have millions of people and millions of stocks and wanted to simulate this "stock market" at 100Hz on a TI-83?

I dunno. You can make up stories.

Re: Matrices and Graph

#44

Post by TivadarDanka »

duckqlz wrote: post #3

Does anyone know how the graph illustrations were created? I have driven myself mad with tikz and dot .

Author here. I rendered (most of the) equations with QuickLaTeX (https://quicklatex.com/), and put the illustrations together in Inkscape by hand (https://inkscape.org/).

Re: Matrices and Graph

#46

Post by dustingetz »

algas wrote: post #42
dpflan wrote: post #6

Earlier quoted context omitted.

I don't know how proof heavy this course is and it is not specialized on graphs: "Computational Science and Engineering I" -- Gilbert Strang (MIT) https://ocw.mit.edu/courses/18-085-computational-science-and...

The textbook for this course is probably one of the finest Strang ever wrote. It's also worth looking at the original edition of the textbook, which was called "Introduction to Applied Mathematics" and has a much more thorough treatment of the parallels between matrices, graphs, and differential operators, and their use in optimization problems. CSE is much more practical for solving actual scientific computing probl…

oh my, thanks! Ed 1 is the exact book i wanted

Re: Matrices and Graph

#47

Post by dpflan »

dustingetz wrote: post #46
algas wrote: post #42

Earlier quoted context omitted.

The textbook for this course is probably one of the finest Strang ever wrote. It's also worth looking at the original edition of the textbook, which was called "Introduction to Applied Mathematics" and has a much more thorough treatment of the parallels between matrices, graphs, and differential operators, and their use in optimization problems. CSE is much more practical for solving actual scientific computing probl…

oh my, thanks! Ed 1 is the exact book i wanted

Were you able to find a copy? I found this on archive.org -- https://archive.org/details/introductiontoap0000stra/page/n9...
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