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Matrices can be your friends (2002)

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Re: Matrices can be your friends (2002)

#91

Post by Ono-Sendai »

nyrikki wrote: post #81
Ono-Sendai wrote: post #45

Earlier quoted context omitted.

I think you are confused about what 'simply connected' means. A 3x3 matrix can represent any rotation. Also from a given rotation there is a path through the space of rotations to any other rotation. It's just that some paths can't be smoothly mapped to some other paths.

SO(3) contains all of the orthogonal 3x3 matrices of determinant 1. If you are dealing with rigid bodies rotated though the origin like with the product of linear translations you can avoid the problem. At least with an orthonormal basis R^3 with an orthogonal real valued 3x3 matrix real entries which, where the product of it with its transpose produces the identity matrix and with determinant 1 But as soon as you ar…

The belt trick is about smoothly mapping paths through SO(3) to each other, not about if any point in SO(3) is reachable from any other point. I think you are confusing the notion of 'simple-connected' with just 'connected'.

Re: Matrices can be your friends (2002)

#92

Post by chris_wot »

recursive wrote: post #66
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…

I took a linear algebra class, as well as many others. It didn't work. Most math classes I've taken granted me some kind of intuition for the subject material. Like I could understand the concept independent from the name of the thing. In linear algebra, it was all a series of arbitrary facts without reason for existing. I memorized them for the final exam, and probably forgot them all the next day, as they weren't a…

You might like to try this website that explains it quite intuitively:

https://betterexplained.com/articles/matrix-multiplication/

Re: Matrices can be your friends (2002)

#93

Post by nerdponx »

knappa wrote: post #30
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.

I assume they're talking about how when multiplying by a matrix by a vector, it's more natural to think of the matrix as a collection of columns than as a collection of rows, and to think of the vector as a column vector.

That layout is a nearly universal convention in applied practice such as statistics. Readers would be very very confused if you flipped it the other way.

The irony is that "programmers" are much more divided on this than statisticians are.

Re: Matrices can be your friends (2002)

#94

Post by jbn »

getnormality wrote: post #73
recursive wrote: post #66

Earlier quoted context omitted.

I took a linear algebra class, as well as many others. It didn't work. Most math classes I've taken granted me some kind of intuition for the subject material. Like I could understand the concept independent from the name of the thing. In linear algebra, it was all a series of arbitrary facts without reason for existing. I memorized them for the final exam, and probably forgot them all the next day, as they weren't a…

I'm an applied math PhD who thinks linear algebra is the best thing ever, and it's the nuts and bolts of modern AI, so for fun and profit I'll attempt a quick cheat sheet. To manage expectations, this won't be very satisfying by itself. You have to do a lot of exercises for this stuff to become second nature. But hopefully it at least imparts a sense that the topic is conceptually meaningful and not just a profusion…

Funny you mention shearing in the last paragraph. Transvection were such a revelation to me in undergrad math.

Re: Matrices can be your friends (2002)

#95

Post by larodi »

shiandow wrote: post #3

I don't think there's any mathematical reason to lay out the elements in memory that way. Sure given no context I would probably use i = row + n col as index, but it doesn't really matter much me. If I had to pick between a matrix being a row of vectors or a column of covectors, I'd pick the latter. And M[i][j] should be the element in row i column j, which is nonnegotiable.

He lost me at

> Well, if we neglect the translation part (the bottom row), then the pure rotation part simply describes the new location of the points on the cube:

Even though im very good at visual with say complex graphs. I really didn’t get where the ‘simply’ part really comes from..

Re: Matrices can be your friends (2002)

#96

Post by larodi »

gus_massa wrote: post #7
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.

Mathematician here. I never heard that. (In many branches the idea is that you care about the abstract linear transformation and properties instead of the dirty coefficients that depend on the specific base. I don't expect a mathematician to have an strong opinion on the order. All are equivalent via isomorphism.)

Thank u so much brother.

Re: Matrices can be your friends (2002)

#97

Post by larodi »

zaik wrote: post #59
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.

Recently graduated math student here. The definition of the "vec" operator which turns a matrix into a vector works like this, stacking up columns rather than rows. https://en.wikipedia.org/wiki/Vectorization_(mathematics)

Really wonder who in his right mind, who can also read a book, would decide to do so in order to enumerate the sequences of items…

Re: Matrices can be your friends (2002)

#98

Post by imtringued »

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…

Rotations:

SO(3) is the special orthogonal group for 3 dimensions, which is the set of rotation matrices. Rotation matrices have the determinant 1, which means that they keep the length of a vector the same (forcing it to change direction). Orthogonal group means that the rotation R times its transpose R^T yields the identity matrix aka R^T R=RR^T=I. This means that rotations are trivially reversible. You can obtain rotation matrices either by explicit use of cosine/sine along your desired axis or you can use the Rodrigues rotation formula to describe a rotation along an axis with an angle.

so(3) is the Lie algebra to the Lie group SO(3). The capitalization here is critical. In particular, so(3) is the set of skew symmetric matrices. 3 by 3 skew-symmetric matrices essentially describe the cross product a x b for a constant a. Given a skew symmetric matrix and a rotation angle theta you can use the exponential function to map so(3) to SO(3) through e^(theta S). Note that the exponential is a general mapping from a Lie algebra to its Lie group.

Transformations:

SE(3) is the special euclidean group for 3 dimensions. Its structure consists of T = [[R, t], [0, 1]].

if R = I, then T is a pure translation by t. If R != I and t = 0, then it is a pure rotation.

When you have both a rotation R and a translation t transformation then translating first and rotating second will give you T, but rotating first and translating second will give you T' = [[R, Rt], [0, 1]].

se(3) is the Lie algebra to the Lie group SE(3). One way to represent so(3) is through twists.

Revolute joints are represented by a vector Xi = [ omega, q cross-product omega] where omega is a vector that represents the parameters of a skew symmetric matrix and q is a 3d point on the axis of rotation.

Prismatic joints are represented by Xi = [0, v] where v is pointing in the direction of the movement. The transformation of a twist can be obtained as usual through the exponential function: exp(theta Xi)=T.

https://en.wikipedia.org/wiki/3D_rotation_group

https://en.wikipedia.org/wiki/Orthogonal_group

https://en.wikipedia.org/wiki/Euclidean_group

https://en.wikipedia.org/wiki/Screw_theory

Re: Matrices can be your friends (2002)

#99

Post by imtringued »

Merad wrote: post #64
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…

> Anyone who has taken linear algebra should know that [...] My university level linear algebra class didn't touch practical applications at all, which was frustrating to me because I knew that it could be very useful to some background doing hobbyist game dev. I still wish I had a better understanding of the use cases for things like eigenvectors/values.

The most obvious application for eigenvalues is that if you find the eigendecomposition A=QλQ^{-1}, where λ=diag(eigenvalues) you can cheaply obtain the results of A^n = QλQ^{-1} ... QλQ^{-1} ... QλQ^{-1} = Qλ^nQ^{-1}. A less likely use is for inverting A or solving a linear system of equations with it.

Re: Matrices can be your friends (2002)

#100

Post by srean »

blackbear_ wrote: post #26
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…

> The fundamental truth is that matrices represent linear transformations, and all of linear algebra is developed in terms of linear transformations...

Not at all. It is most definitely not a fundamental truth. I think you are conflating matrix multiplication with matrices, as if that's the only operation that one does on matrices.

Matrices are a way to organize data and what operations you define over the matrices define what it represents.

There are other useful operations on matrices, for example Hadamard products, Schur products, Kronecker product, face split products of matrices that finds use in Physics and Engineering (antenna design for one) where the operation does not represent a linear transformation. They can form an algebra and/or algebraic structures different from the very familiar linear algebraic one.

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