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Matrices can be your friends (2002)

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Re: Matrices can be your friends (2002)

#21

Post by voidUpdate »

foofoo12 wrote: post #19
omnicognate wrote: post #13

Earlier quoted context omitted.

A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist. I'd take issue with his "most programmers are visual…

> most programmers are visual thinkers I remember reading that there's a link between aphantasia (inability to visualize) and being on the spectrum. Being an armchair psychologist expert with decades of experience, I can say with absolute certainty that a lot of programmers are NOT visual thinkers.

Do you have anything I can read about that? I'm definitely on the spectrum and have whatever the opposite of aphantasia is, I can see things very clearly in my head

Re: Matrices can be your friends (2002)

#22

Post by kevindamm »

dgacmu wrote: post #17
zelphirkalt wrote: post #15

Earlier quoted context omitted.

When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there w…

You can do rotation with a 3x3 matrix. The first lecture was using a 4x4 matrix because you can use it for a more general set of transformations, including affine transforms (think: translating an object by moving it in a particular direction). Since you can combine a series of matrix multiplications by just pre-multiplying the matrix, this sets you up for doing a very efficient "move, scale, rotate" of an object usi…

The first time I learned it was from a book by LaMothe in the 90s and it starts with your demonstration of 3D matrix transforms, then goes "ha! gimbal lock" then shows 4D transforms and the extension to projection transforms, and from there you just have an abstraction of your world coordinate transform and your camera transform(s) and most everything else becomes vectors. I think it's probably the best way to teach it, with some 2D work leading into it as you suggest. It also sets up well for how most modern game dev platforms deal with coordinates.

Re: Matrices can be your friends (2002)

#23

Post by scarmig »

zelphirkalt wrote: post #15
getnormality wrote: post #6

People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! Anyone who has taken linear algebra should know that (1) a rotation is a linear operation, (2) the result of a linear operation is calculated with matrix…

When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there w…

The mathematical term for the four by four matrices you were looking at is "quaternion" (I.e. you were looking at a set of four by four matrices isomorphic to the unit quaternions).

Why use quaternions at all, when three by three matrices can also represent rotations? Three by three matrices contain lots of redundant information beyond rotation, and multiplying quaternions requires fewer scalar additions and multiplications than multiplying three by three matrices. So it is cheaper to compose rotations. It also avoids singularities (gimbal lock).

Re: Matrices can be your friends (2002)

#24

Post by ndriscoll »

card_zero wrote: post #18
ljlolel wrote: post #16

Earlier quoted context omitted.

Take linear algebra

OK, now what?

A matrix is just a list of where a linear map sends each basis element (the nth column of a matrix is the output vector for the nth input basis vector). Lots of things are linear (e.g. scaling, rotating, differentiating, integrating, projecting, and any weighted sums of these things). Lots of other things are approximately linear locally (the derivative if it exists is the best linear approximation. i.e. the best matrix to approximate a more general function), and e.g. knowing the linear behavior near a fixed point can tell you a lot about even nonlinear systems.

Re: Matrices can be your friends (2002)

#25

Post by morcus »

Viliam1234 wrote: post #5

> Mathematicians like to see their matrices laid out on paper this way (with the array indices increasing down the columns instead of across the rows as a programmer would usually write them). Could a mathematician please confirm of disconfirm this? I think that different branches of mathematics have different rules about this, which is why careful writers make it explicit.

Most fields of math that use matrices don't number each element of the matrix separately, and if they do there will usually be two subscripts (one for the row number and one for the column number).

Generally, matrices would be thought in terms of the vectors that make up each row or column.

Re: Matrices can be your friends (2002)

#26

Post by blackbear_ »

zkmon wrote: post #12

There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…

The fundamental truth is that matrices represent linear transformations, and all of linear algebra is developed in terms of linear transformations rather than just grid of numbers. It all becomes much clearer when you let go of the tabular representation and study the original intentions that motivated the operations you do on matrices.

My appreciation for the subject grew considerably after working through the book "Linear Algebra done right" by Axler https://linear.axler.net

Re: Matrices can be your friends (2002)

#27

Post by yoda222 »

Viliam1234 wrote: post #5

> Mathematicians like to see their matrices laid out on paper this way (with the array indices increasing down the columns instead of across the rows as a programmer would usually write them). Could a mathematician please confirm of disconfirm this? I think that different branches of mathematics have different rules about this, which is why careful writers make it explicit.

Not a mathematician, just an engineer that used matrix a lot (and even worked for MathWorks at one point), I would say that most mathematicians don't care. Matrix is 2D, they don't have a good way to be laid out in 1D (which is what is done here, by giving them linear indices). They should not be represented in 1D.

The only type of mathematicians that actually care are: - the one that use software where using one or the other and the "incorrect" algorithm may impact the performance significantly. Or worse, the one that would use software that don't use the same arbitrary choice (column major vs row major). And when I say that they care, it's probably a pain for them to think about it. - the one that write these kind of software (they may describe themselves as software engineer, but some may still call themselves mathematicians, applied mathematicians, or other things like that).

Now maybe what the author wanted to say is that some language "favored by mathematician" (Fortran, MATLAB, Julia, R) are column major, while language "favored by computer scientist" (C, C++) are row major

Re: Matrices can be your friends (2002)

#28

Post by aqme28 »

zelphirkalt wrote: post #15
getnormality wrote: post #6

People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! Anyone who has taken linear algebra should know that (1) a rotation is a linear operation, (2) the result of a linear operation is calculated with matrix…

When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there w…

Yeah you need to build up the understanding so that you can re-derive those matrices as needed (it's mostly just basic trigonometry). If you can't, that means a failure of your lecturer or a failure in your studying.

Re: Matrices can be your friends (2002)

#29

Post by nyrikki »

zkmon wrote: post #12

There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…

the set of all matrices of a fixed size are a vector space because matrix addition and scalar multiplication are well-defined and follow all vector space axioms.

But be careful of the map–territory relation.

If you can find a model that is a vector space, that you can extend to an Inner product space and extend that to a Hilbert space; nice things happen.

Really the amazing part is finding a map (model) that works within the superpowers of algorithms, which often depends upon finding many to one reductions.

Get stuck with a hay in the haystack problem and math as we know it now can be intractable.

Vector spaces are nice and you can map them to abstract algebra, categories, or topos and see why.

I encourage you to dig into the above.

Re: Matrices can be your friends (2002)

#30

Post by knappa »

Viliam1234 wrote: post #5

> Mathematicians like to see their matrices laid out on paper this way (with the array indices increasing down the columns instead of across the rows as a programmer would usually write them). Could a mathematician please confirm of disconfirm this? I think that different branches of mathematics have different rules about this, which is why careful writers make it explicit.

I'm a mathematician. It's kind of a strange statement since, if we are talking about a matrix, it has two indices not one. Even if we do flatten the matrix to a vector, rows then columns are an almost universal ordering of those two indices and the natural lexicographic ordering would stride down the rows.
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