structures:matrix:applications
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structures:matrix:applications [2025/03/03 10:29] simon |
structures:matrix:applications [2025/03/03 10:52] (current) simon [working with matrices] |
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| Maybe this whole discussion is hinting why HSC Physics is kind of lacking. The teachers cannot even depend on students knowing calculus, maths is not a co-requisite, and yet even extension 2 maths only gives a taste of some of the language required to build modern physics. But it is not just Physics. The mathematical structures I will give an extremely abbreviated overview of are fundamental to analysis of all kinds not just engineering, and to computation and design. | Maybe this whole discussion is hinting why HSC Physics is kind of lacking. The teachers cannot even depend on students knowing calculus, maths is not a co-requisite, and yet even extension 2 maths only gives a taste of some of the language required to build modern physics. But it is not just Physics. The mathematical structures I will give an extremely abbreviated overview of are fundamental to analysis of all kinds not just engineering, and to computation and design. | ||
| - | Matrices are everywhere. They can be operators transforming vectors or shapes in cartesian space. GPUs, the 'graphics cards' that run the screens on your devices are just very parallel matrix calculators, they do matrix operations fast and do many at the same time. Photoshop or whatever graphics applications you use do matrix operations constantly. Solving a set of [[#linear algebra|linear simultaneous equations]], like we do in year 8 in high school, is basically the original motivation for matrices. That problem has been an inspiration for maths going way way back, one of the earliest mathematics --- but not in europe, europe focussed on formal geometry instead and eventually imported this stuff. The coefficients form a matrix where each row is an equation and the steps we do by hand are matrix operations. A wide range of practical problems in business and [[#planning]] are, if you frame them that right way, combinations of those kinds of relationships and we are looking for the best outcome, the highest or lowest overall value. This is called [[#optimisation]], and in its simplest form it is something we study in high school. | + | Matrices are everywhere. They can be operators transforming vectors or shapes in cartesian space. GPUs, the 'graphics cards' that run the screens on your devices are just very parallel matrix calculators, they do matrix operations fast and do many at the same time. Photoshop or whatever graphics applications you use do matrix operations constantly. Solving a set of [[#linear algebra|linear simultaneous equations]], like we do in year 8 in high school, is basically the original motivation for matrices. That problem has been an inspiration for maths going way way back, one of the earliest mathematics --- but not in europe, europe focussed on formal geometry instead and eventually imported algebra and the core numerical algorithms. Matrices came later. The coefficients form a matrix where each row is an equation and the steps we do by hand are [[#linear algebra|matrix operations]]. A wide range of practical problems in business and [[#planning]] are, if you frame them that right way, combinations of those kinds of relationships and we are looking for the best outcome, the highest or lowest overall value. This is called [[#optimisation]], and in its simplest form it is something we study in high school. |
| - | A supercomputer, with thousands of those matrix calculating GPUs repurposed for more general matrix work, runs simulations with varied assumptions using matrix operations and finds the best or average of the ones it tries by iterating, by stepping in little jumps through time. But more detail makes this very much a longer calculation, a supercomputer cannot manage to cope when we try more detail. Now we use trained Neural Nets and are just right now getting Quantum Computers going. But both of these are much more sophisticated and intense ... matrix calculations!!! This time much further abstracted from step by step logic. | + | A supercomputer, with thousands of those matrix calculating GPUs repurposed for more general matrix work, runs simulations with varied assumptions using matrix operations and finds the best or average of the ones it tries by iterating, by stepping in little jumps through time. But more detail makes this very much a longer calculation, a supercomputer cannot manage to cope when we try more detail. Now we use trained Neural Nets and are just right now getting Quantum Computers going. But both of these are much more sophisticated and intense ... matrix calculations!!! This time much further abstracted from step by step logic, they can no longer be described as made up of parts like the examples below, or as representing this or that transformation in some other context. |
| ====working with matrices==== | ====working with matrices==== | ||
| - | So far we know a [[discussion#matrices|matrix]] is a table of [[discussion#scalar]]s (often Real Numbers), rows and columns sometimes representing data or perhaps representing a list of [[discussion#vectors]]. But perhaps it is just a matrix in its own right. I have mentioned we can add them and multiply them by a value from their scalars, and that they can represent the transformations we have studied in cartesian [[discussion#space]] (rotation, stretching etc). I also mentioned that [[discussion#Complex Numbers]] can fit in here in an intriguing way. So here are some details. | + | So far we know a [[discussion#matrices|matrix]] is a table of [[discussion#scalar]]s (often Real Numbers), rows and columns sometimes representing data or perhaps representing a list of [[discussion#vectors]]. But perhaps it is just a matrix in its own right. I have mentioned we can add them and multiply them by a value from their scalars, and that they can represent the transformations we have studied in cartesian [[discussion#space]] (rotation, stretching etc). I also mentioned that [[discussion#Complex Numbers]] can fit in here in [[discussion#complex-as-matrix|an intriguing way]]. So here are some details. |
| Any matrix can be multiplied by its type of scalar, multiply each element individually: | Any matrix can be multiplied by its type of scalar, multiply each element individually: | ||
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