Exploring Matrices: Forms, Dimensions and Mechanisms

In summary, the conversation discusses the use of matrices in various fields, such as programming and 3D graphics. Matrices are used for coordinate transformations and can be expressed as a linear equation system. One interesting aspect of matrices is that their multiplication is noncommutative, leading to different results depending on the order of multiplication.
  • #1
drag
Science Advisor
1,105
1
Greetings !

Can anyone please explain to me, preferably with as
little math and as much theory as possible , how
matrices work in terms of forms and dimensions and what's
the relevant mechanism ?

I mean, I used to program Star Trek spaceships for a computer
game I once made using matrices. But, nobody ever explained
to me how and why it works, those were just meaningless tables
with sprecific relevant operators to me. I also know the
whole addition/multiplication/other operations stuff(though
I can hardly remember it now), but not how and why it works for geometrical forms in various dimensions and stuff.

Thanks ! :smile:

Live long and prosper.
 
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  • #3
OK so maybe this is a response for a restricted area of matices use but here goes.
I program in OpenGL and I think the other 3D programming stuff has the same principles. To make a 3D scene you have to use coordinate transformation. These transformations (translation, rotation, scaling) can be written as a linear sistem between the coordinates in the old reference system, and the coordinates in the new reference system. Any linear equation system can be expressed in a matriceal form. So this is one of the reasons you use matrices. Also basic matrix operations are easy to program.
 
  • #4
Or, in other words ...
You can do all the world transformation without matrices, it is only that matrices is a convinient way to save points, data, and transformations.
(i learned that while trying to learn Direct3D)
 
  • #5
One of the most interesting things about matrices are than, due to the method of multiplication A * B is not necessarily the same as B * A.
 
  • #6
One of the most interesting things about matrices are than, due to the method of multiplication A * B is not necessarily the same as B * A.

Yep, Matrix multiplication is noncommutative and I know some people who failed exams because of it (even though the teacher had mentioned many times).
 
  • #7
Originally posted by plus
One of the most interesting things about matrices are than, due to the method of multiplication A * B is not necessarily the same as B * A.

true, true...
that's why in OpenGL a rotation, followed by a translation doesn't have the same result as a translation followed by a rotation. To do that you have to make some matrix multiplication and since it's not commutative the results are dependent of the order in which you multiply.
 

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is a fundamental tool in mathematics and is used to represent and manipulate data in various fields such as physics, engineering, and computer science.

2. What are the different forms of matrices?

There are several forms of matrices, including square matrices, rectangular matrices, row matrices, column matrices, and zero matrices. These forms differ based on the number of rows and columns they have, as well as their specific properties and uses.

3. What is the importance of dimensions in matrices?

The dimensions of a matrix refer to the number of rows and columns it has. The dimensions play a crucial role in determining the operations that can be performed on the matrix, as well as its rank, determinant, and other properties. In addition, the dimensions of matrices are used to define the size of vectors and other mathematical objects.

4. How are matrices used in real-world applications?

Matrices have a wide range of applications in various fields. For example, they are used in data analysis and machine learning to represent and process large datasets, in computer graphics to create 3D objects and animations, and in economics to model and solve systems of equations. They are also used in physics, engineering, and other sciences to represent physical quantities and equations.

5. What are the mechanisms involved in matrix operations?

Matrix operations involve various mechanisms, such as addition, subtraction, multiplication, and inversion. These operations follow specific rules and algorithms to manipulate the elements of matrices and produce new matrices with different properties. These mechanisms are essential for solving systems of equations, finding determinants, and performing other mathematical operations involving matrices.

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