Understanding matrixes

[grayson]

I am not going to specify my age here. But I am ahead of my time. I finally understand most of calculus and now want to get into some other complicated stuff. The only problem is that I have a short attention span. I need something I can read and something that keeps it interesting to understand matrixes or linear algebra. Can anyone help me?

[Engineeer]

Understanding duress or the need for pain in the body is the false sense of hope it gives the ___

Understanding the emotion of hubris is derived from the coming of it's opposite. 

Edited November 15, 2023 by Engineeer

[Sensei]

On 8/20/2023 at 9:50 PM, grayson said:

I am not going to specify my age here. But I am ahead of my time.

Why do you write such nonsense?

On 8/20/2023 at 9:50 PM, grayson said:

I finally understand most of calculus and now want to get into some other complicated stuff. The only problem is that I have a short attention span. I need something I can read and something that keeps it interesting to understand matrixes or linear algebra. Can anyone help me?

Matrix operations are nothing more than operations on multiple variables simultaneously/parallely.

 

If you have matrix 2x2, you have 4 equations.

If you have matrix 3x3, you have 9 equations.

If you have matrix 4x4, you have 16 equations. etc.

 

The operation between vector with n-th element and matrix which has n-th x n-th element, can be expressed (for somebody reading it) as an array ("matrix") of equations.

 

For example (the simplest case), the scale matrix will have a format such as:

sx, 0, 0,

0, sy, 0,

0, 0, sz

https://en.wikipedia.org/wiki/Scaling_(geometry)

https://www.google.com/search?q=scale+matrix

 

How to perform operation with a such matrix?

Multiple columns by vector [x,y,z] element and add them together in the same row.. e.g.

x' = sx * x + 0 * y + 0 * z;

y' = 0 * x + sy * y + 0 * z;

z' = 0 * x + 0 * y + sz * z;

Because matrix had 0 in some columns and rows (most of them!), this can be much simplified without matrix multiplication to just:

x' = sx * x

y' = sy * y

z' = sz * z

From 9 equations (3x3 matrix with 3x vector elements, 9) we can have 3.

From 16 equations (4x4 matrix with 4x vector elements, 16) we can have 4.

 

For rotation it is not so simple. E.g. rotation in Y axis:

Matrix 3x3 will have format e.g.

cos( angle), 0, -sin( angle ),

0, 1, 0,

sin( angle ), 0, cos( angle ),

Or something like that (+/- sinuses and cosines)

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

 

For translation you need 4x4 matrix with format:

1, 0, 0, tx,

0, 1, 0, ty,

0, 0, 1, tz,

0, 0, 0, 1,

https://www.google.com/search?q=translation+matrix

 

Matrix operations are 1) easy for computers to perform in parallel 2) slow down the whole process (during plain scaling and/or translation) 3) speed up operations if they were complex are they were merged together 4) make it more readable for humans who needs to review them.

The matrices can be "multiplied" by other matrices, yielding matrices that can be used for the final vector transformation. Which can simplify the entire vector transformation process.

https://www.google.com/search?q=matrix+combine

https://www.google.com/search?q=matrix+merge

 

MMX means Matrix Math Extensions or Multi Media eXtension. Pentium CPU ~ 26 years ago.

SSE means Streaming SIMD Extensions.

They are basically operations on matrices and vectors. Multiply 2, 3 or 4 elements in an array and/or add them together in the one CPU operation.

 

If you write things the right way, MMX and/or SSE/SSE2, etc. will be automatically activated by the compiler and used, and you will be able to increase the performance of your code... or you can write your own MMX/SSE/SSE2+/AVX CPU code..

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

(AVE = "Advanced Vector Extensions" > SSE > MMX)

i.e. AVE is SSE on "steroids".. and SSE is MMX on steroids...

..now depreciated by GPUs..

 

Edited November 15, 2023 by Sensei

[Engineeer]

It's about understanding mistakes 

[studiot]

On 8/20/2023 at 8:50 PM, grayson said:

I am not going to specify my age here. But I am ahead of my time. I finally understand most of calculus and now want to get into some other complicated stuff. The only problem is that I have a short attention span. I need something I can read and something that keeps it interesting to understand matrixes or linear algebra. Can anyone help me?

Noting your comment about attention span here is my short form answer.

 

Matrices refer to when a bunch of things that have some sort of connection between them so that,  taken as a whole they form something that is greater than their individual parts.

The english word comes from the latin prefix 'matri' which refers to female matters and more particularly the latin word 'matrice' which refers to the womb.

 

In English, the word matrix has been used to indicate this connection in many wide ranging places .

For instance in education, matriculation indicates a connection between the subjects studied.
Such connections may be shown in a data table, also called a matrix.

Many subjects, including formal logic employ data tables.
These types of matrix may or may not have a connection to linear algebra.

In Materials Science, Chemistry, Engineering  and related disciplines a matrix is formed when a large bunch of atoms are connected or linked (by chemical bonds) to form a continuous material.
Other terms for this are substrate.

An example would be the resin matrix in fibre reinforced plastic.

 

In linear algebra a matrix is a data table in which all the entries are numbers or functions which have definite numerical values.
Another word for this is an array of numbers.
The connection between these numbers follows the rules of linear algebra.