[HN Gopher] 2000 Years of Matrix Multiplication
___________________________________________________________________
2000 Years of Matrix Multiplication
Author : nothrowaways
Score : 168 points
Date : 2023-02-04 10:51 UTC (12 hours ago)
(HTM) web link (mathshistory.st-andrews.ac.uk)
(TXT) w3m dump (mathshistory.st-andrews.ac.uk)
| meken wrote:
| Determinants always fascinated me, partly (completely?) because
| of the cool sounding name. I always wanted to get intuition for
| them because their formula is so simple, which has always been
| deeply unsatisfying to me to just take on its face.
|
| This article got me searching for a 3blue1brown video on
| determinants and now my mind is absolutely blown!
|
| https://youtu.be/Ip3X9LOh2dk
| lambdaxymox wrote:
| One of the great parts about linear algebra is that there is
| almost always a simple geometric idea underneath.
|
| The geometric picture underneath is one of the things that
| keeps me in awe of the subject despite its seeming simplicity,
| and I keep getting something out of it every time I come back
| to it. It's a bummer since finite-dimensional linear algebra is
| one of a handful of mathematics topics where one can answer all
| the questions posed at the beginning of a course in it by the
| end of a course in it, so it is a pretty self-contained topic.
|
| After learning e.g. exterior algebra (differential forms),
| Clifford algebra (geometric algebra in these parts), and so on,
| the geometric picture of the determinant as the size of an
| oriented volume makes deriving the algebraic formula super
| duper slick. Like in Clifford algebra, the formula can be
| proven in two or three lines. It's unfortunate that it seems
| like e.g. exterior algebra never get introduced sooner in the
| pedagogy of linear algebra or multivariable calculus because
| when used right they make the underlying ideas shine through
| beautifully. It's a bummer since exterior algebra is much
| simpler than it looks, though like many things in mathematics,
| it's takes a lot of work to make that simple idea rigorous. But
| unfortunately algebra in general given it's abstract nature can
| absolutely lobotomize the real deal geometric ideas underneath
| a lot of this stuff when used poorly.
| flebron wrote:
| The trifecta of: * There's almost always a
| simple geometric intuition, and low-dimensional intuition can
| get you quite far even in high dimensional cases. * You
| can surprisingly often get by with closing your eyes and
| saying "my problem is linear" three times. See: All of neural
| networks. * Linear problems have practically all nice
| properties you could ever ask of any function.
|
| Has made linear algebra by far the most bang/buck mathematics
| topic I've studied in my life. Close behind is asymptotic
| analysis.
| edflsafoiewq wrote:
| A shocking number of people are apparently taught the
| determinant without learning the geometric interpretation. It's
| educational malpractice.
| IIAOPSW wrote:
| watch his fourier trick. you're welcome
| ndriscoll wrote:
| It's unfortunate that more theory makes determinants really
| intuitive, but most people don't get there. The upshot is that
| there's a thing called the exterior algebra which is relatively
| simple to define and calculate with, and essentially encodes
| the notions of areas, volumes, 4-volumes, etc. over a space.
| Every linear map on a space uniquely specifies a map on the
| exterior algebra (in fact this is a functor, which makes
| calculations easier), and the determinant of a linear map is
| then just the corresponding exterior algebra map evaluated on
| the unit n-volume. The minors are the exterior algebra map
| evaluated on the unit volume for various subspaces.
|
| The rules for the exterior algebra and the fact that this is a
| functor let you learn a couple simple rules to simplify
| expressions into a standard form, and then you just apply those
| rules mechanically and don't have to remember minus signs or
| what multiplies with what. It becomes a process that requires
| no thought.
|
| It's sort of like how once you learn how to deal with complex
| exponentials, you can forget pretty much every rule from
| trigonometry, making it entirely pointless to memorize those
| rules.
| green_on_black wrote:
| It's unfortunate that lower level "math" is so banally taught
| (i.e. calc and below), at least in the US.
| dahart wrote:
| It's interesting and cool to see how long it took, and how many
| people it took, for the abstraction of matrices and matrix
| algebra to develop (in addition to how well-traveled the road was
| before the people we've attributed to and named some of these
| techniques after). What this makes me wonder is: what
| abstractions might be in progress today but incomplete? Maybe
| it's a fun thought experiment and there's no way to know because
| the ideas haven't occurred to anyone, or maybe some people
| already do have an idea but won't ultimately be remembered for it
| when our progeny is able to explore it a little more deeply and
| explain it a little more clearly. Or, I don't know, maybe the
| period of history where some things get forgotten is now over?
| jiggawatts wrote:
| > What this makes me wonder is: what abstractions might be in
| progress today but incomplete?
|
| Category Theory, especially how it relates to programming
| language and API design.
|
| We're building pyramids like the Egyptians did, and we're
| impressed without ourselves and our grand achievements just
| like they were.
|
| The Egyptians built a wonder of the world before the invention
| of the wheel!
|
| That's what programming feels line to me. We're well paid
| labourers chiselling away at stone with crude tools. We're
| convinced ourselves that we live in the future because we have
| copper tools instead of just wood and stone.
|
| The most amazing thing about the computer revolution is that it
| hasn't happened yet.
| jtimdwyer wrote:
| If you have not, you may be interested in reading Thomas Kuhn's
| book -- The Structure of Scientific Revolutions.
| msravi wrote:
| Several methods of solving linear equations of several unknowns
| are described in Hindu literature, dating from about 200CE
| onwards. In particular, Brahmagupta's method seems very similar
| to Gaussian elimination[1].
|
| And then there are solutions to quadratics, cubics and
| biquadratics, pell's equations, etc.
|
| 1. https://archive.org/details/history-of-hindu-
| mathematics-2-b...
| joe__f wrote:
| Pretty mind boggling that there were Babylonians solving
| simultaneous equations, and people knew Gaussian elimination in
| 3rd century BC China. I wonder if the simultaneous equations were
| actually used to guide any decisions made by Babylonian
| government
| thechao wrote:
| There were Sumerians solving differential equations at least
| 3700 years ago. Irving Finkel has a wonderful book about this:
| "The Ark Before Noah".
| selimthegrim wrote:
| Is this the part in the book where they calculate the shape?
| whatshisface wrote:
| Difference equations, not differential equations.
| theteapot wrote:
| I have a stupid question. I still don't really fully understand
| why matrix multiplication came to be row by column. I mean you
| could define things as row by together with the transpose
| operator right? So why is row by column so obviously the right
| way to define multiplication?
| tptacek wrote:
| I'm bad at math, and know some linear algebra only because it's
| useful in cryptanalysis, but the way I keep this in my head is
| that matrix multiplication, mechanically, is just repeated
| application of matrix-vector multiplication; you're just doing
| multiple columns, not just one. And then, matrix-column vector
| multiplication is linear function application, right?
|
| And you can do column-times-row too, right? It's something like
| the sum of multiple outer products?
|
| (These question marks are all genuine! I probably have most of
| this wrong!)
| ecnahc515 wrote:
| It's probably related to how we read and write. Top down, left
| to right. Rows are the entries, columns are attributes. Your
| list will potentially grow forever, whereas the attributes are
| typically fixed. Your width is typically fixed, but you can
| continue writing top down and continue on a new sheet/etc, as
| you add more rows. It just makes sense with how we typically
| organize information on pen/paper.
| 082349872349872 wrote:
| cf http://conal.net/blog/posts/reimagining-matrices
| 22022022 wrote:
| cf https://mbuliga.github.io/quinegraphs/puresee.html
| 082349872349872 wrote:
| > _the emergence of the beta rewrite in lambda calculus from
| the shuffle rewrite and relations to the commutativity of the
| addition of vectors in the tangent space of a manifold_
| (https://mbuliga.github.io/novo/presentation.html)
|
| piques my interest; can you recommend any presentations of
| this work that might be suitable to someone without a
| background in knot theory?
| 22022022 wrote:
| Hi thanks for the interest, for the pure algebraic part a
| good entry is this (and references therein)
| https://arxiv.org/abs/2110.08178
|
| The emergence of the beta rewrite is explained if you try
| to decorate the graphs from the first figure in the
| "Emergent rewrites", according to the dictionary in the
| first section, then you pass to the limit.
|
| To see why is that a beta rewrite you have to go to the
| parent page of Pure See and look for the first article
| listed there.
___________________________________________________________________
(page generated 2023-02-04 23:01 UTC)