# What exactly are matrices used for?

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#1
What real world applications are they used in? I understand how they work but what sort of application would they be used for?
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4 years ago
#2
(Original post by AishaGirl)
What real world applications are they used in? I understand how they work but what sort of application would they be used for?
Matrices are used all over maths but for uses outside maths:

"Data that can be organized in rows and columns can be represented as a matrix. In computer science, matrices are used in the projection of three dimensional images into two dimensional screens. Scholastic matrices are used by page rank algorithms*. Cryptography is also implemented using matrices.

In physics, matrices are applied in optics, quantum mechanics and electrical circuits. Matrices can also be used to represent real-world data like the traits or habits of a population. Matrices are also applied in economics to study stock market trends and to optimize profit and minimize loss, in chemistry to find quantities in a chemical solution, and in genetics to work out the selection process."

*Page rank algorithms are used by internet search engines e.g. Google.
2
4 years ago
#3
(Original post by AishaGirl)
What real world applications are they used in? I understand how they work but what sort of application would they be used for?
So much! In my first year of undergraduate maths and physics, matrices have cropped up pretty much in all sorts of places. The applications to applied mathematics (i.e. physics) is huge. The rotations that matrices can represent make them excellent for particle physics and looking at rotational symmetries. Other sorts of symmetries in particle physics also make use of matrices. Saying that, a pretty large proportion of physics in general makes use of matrices.

Matrices can be used to solve systems of equations. Solving simultaneous equations with four variables can be tedious enough, but when there is data that introduces many more variables, it'd be a complete mess to do it by hand. Row reductions can be performed on matrices to make this much more simpler. So data handling using matrices is quite a good one. The Jacobean matrix is also quite a useful one when it comes to looking at ordinary differential equations - which themselves have many real-world applications. It's also useful for looking at integrals in more than one dimension where variable transformations are needed - again, multiple integrals themselves have important real world applications (e.g. pressure-volume / gaussian surfaces in physics).

From what I've seen, I'd also say that matrices are just generally very good tools in making much more advanced maths (and mathematical physics) simpler. So a lot of other applications can also be found in how they form incredible forms of machinery that allow for other 'steps' in maths/physics to be made.
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#4
So are matrices used mostly for data management rather than actually solving a specific problem?
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4 years ago
#5
(Original post by AishaGirl)
So are matrices used mostly for data management rather than actually solving a specific problem?
They are also used to solve equations which describe 'real-life' problems.

There is not just one specific use but many ...
0
4 years ago
#6
(Original post by AishaGirl)
So are matrices used mostly for data management rather than actually solving a specific problem?
This is a (small) issue I have with what Notnek quoted. As I see it, there's a distinction between "a rectangular arrangement of data as rows and columns" (which I'd call a 2D-array) and the mathematical concept of a matrix, which has particular operations defined (in particular matrix multiplication, which is quite different from data management). 2D-arrays are used all over the place for what you call data-management, and this is far more common than any use that actually requires you to know anything about matrices.

As a particular instance, images are a common example of something typically represented as a rectangular arrangement of data, but not really having anything to do with matrices.

But anything where you applying a linear transformation to something is typically going to be represented by a matrix. And it's more than just linear problems in practice; we solve a huge number of non-linear problems by assuming that for small changes, we can treat the problem as linear (and then add up the result of lots and lots of small changes).

That said, if I'm completely honest, most people aren't ever going to use matrices once they leave academia. I work in computer graphics and its surprising how many people I work with aren't at all confident with matrix math.
1
4 years ago
#7
When you say 'real world' it depends on whether you mean it in an everyday sense or, generally, 'the world'.

As others have said, matrices are used extensively in STEM subjects.

Those who are doing advanced maths A-level (or whatever it's called... the one with matrices in it) but don't continue on to do anything more technical, will probably not ever make use of matrices in the wider world.
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#8
(Original post by Pessimisterious)
When you say 'real world' it depends on whether you mean it in an everyday sense or, generally, 'the world'.

As others have said, matrices are used extensively in STEM subjects.

Those who are doing advanced maths A-level (or whatever it's called... the one with matrices in it) but don't continue on to do anything more technical, will probably not ever make use of matrices in the wider world.
I mean in the real world with physical items like developing computers, iphones, processors, satellites, sonar, radar etc. I assume there is a lot of math that goes into developing tech, how much of that is dependent on matrices I wonder.
0
4 years ago
#9
They are used for solving equations, making multiple computations faster, storing data.
One of their cooler uses are Markov chains, which are very important in both math and computer science
0
4 years ago
#10
(Original post by AishaGirl)
I mean in the real world with physical items like developing computers, iphones, processors, satellites, sonar, radar etc. I assume there is a lot of math that goes into developing tech, how much of that is dependent on matrices I wonder.
It depends on what you mean by "used". For example, sonar, radar, etc, systems often are designed to use mathematical transforms like the Discrete Fourier Transform, Karhunen-Loeve Transform, z-transform, and so on, and these are mathematically *described* using matrix notation. However, there is no guarantee that they are *implemented* using anything that you would recognise as a matrix - for example a 2D digital image filter can be implemented in hardware, and then it may be computed by multiple reuse of summing circuits of various kinds, with no specific "matrix calculator" as such being present.
0
4 years ago
#11
(Original post by AishaGirl)
I mean in the real world with physical items like developing computers, iphones, processors, satellites, sonar, radar etc. I assume there is a lot of math that goes into developing tech, how much of that is dependent on matrices I wonder.
Absolutely tons.

For example there's information theory, which is almost entirely about matrix manipulation. One aspect is known as coding theory, which is the study of sending information in coded form, in such a way that the information can be decoded even if the data received has been scrambled along the way.

In the real world, most information is sent as binary code. The code has to be sent as a kind of analogue signal, like a radio wave or whatever. If the radio wave is scrambled or goes through some kind of natural interference, the code received may be different form what was sent.

You can use decoding matrices to figure out the correct information that was sent.

For example (and without going into any detailed explanation), have a look at this.

Take the decoding matrix:

and an 'error code'

Now,

Keep in mind that final result: (0,1).

Now imagine a different code was sent to someone. They don't know what information was first sent, but they received the code

This can be checked for errors by using the decoding matrix.

in binary form

This is the same result as was achieved using the (0,1,0,0) error code, and so it can be inferred that received code has a single error in the second digit.

Hence, the received code is decoded to:

This can be double checked by putting the new word through the decoding matrix:

in binary form

The result is (0,0), meaning there is no error in that information, hence (1,1,1,0) is definitely what should have been received.

So to sum things up: The code (1,0,1,0) was received, but the person on the receiving end knows with certainty that it should actually have been (1,1,1,0).

From this I hope you can see how matrices are used in the coding and decoding of information. An error can be detected and decoded from damaged or fuzzy information.

This is absolutely vital in all areas of digital communication. Basically the internet itself and all information transmission relies entirely on these principles.
2
4 years ago
#12
There are tensors that were made in 1969 and can only be used recently. They are many matrices together like a Rubik's cube that are really powerful for data handling and can be used to make games perform amazingly but they require computing power squared proportional to the size of the tensor.
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