In what ways are matrices used in the real world?
In: Mathematics
Matrices are the basis of a lot of the programming calculations. As mentioned in other comments, predictions of weather, robot locations, and other mathematical calculations all use matrices. Matrices can be very complex for a human but a computer can easily complete them. We just see the end result and not what’s going on behind the scenes with the matrix.
One of the coolest used I have seen is to calculate the position the work end of a robot with multiple joints.
Matrices are a form of representing data in a mathematical format
Without going into detail, matrices are used in many things in our everyday life, such as predicting/forecasting the weather, or calculating the statistics of diseases (such as the current pandemic) or even in finding best paths to go from point A to point B on a map
We just don’t get to see them because we have made software that does all the calculations on matrices behind the scenes to give us lines and shapes with pretty colours that we like/prefer to see
Another one: you can do a transition matrix that tracks how people move between groups. For instance, you could have the groups
* healthy but susceptible to a disease
* infected with the disease
* healthy and immune to the disease
* dead
and your matrix would contain the frequencies with which people shift from one group to another each week. If you take your original population and multiply it by the transition matrix over and over, it’ll predict what’s going to happen over time.
Another one: network analysis. If you have some kind of graph like this https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg
…you can write a connectivity matrix where element (ij) equals the number of paths from i to j. Once you have that, the square of the matrix shows you the places you can reach with two jumps, the cube shows you places you can reach with three jumps, and so on.
In circuitry, you can solve a complicated circuit by using node or loop analysis.
Node analysis involves writing an equation for each intersection in the circuit, because the total current at each node has to be zero. (Flow in = flow out.)
Loop analysis involves writing one equation for each closed path in the circuit, because the sum of all the voltage gains/drops around each path has to come out to zero.
Either way, you get a system of equations talking about the currents or voltages in the circuit and if you solve that (take it to reduced row echelon form, for instance) you’ve solved the circuit.
Matrices is used to describe linear function (or relatedly, affine function), and metric tensor, all of which come from linear algebra. Linear algebra is a very useful topic that seep into pretty much everything, because it’s one of the most well-understood math topic. Think about it. Even if you know that your car don’t move at constant speed, you still think of its speed as an useful. Even though we know that trend don’t keep moving, we still try to predict where it will go if it keeps going. Even though we know light don’t go straight, we still treat it as straight line. We understood linear stuff well, and we try to reduce more complicated things to it as our first approximation. Which is why linear algebra is everywhere.
Just 2 examples:
– Statistics: principal component analysis and linear regression are 2 common tool in statistics. They allow you to see how one data can linearly varies with another. Since statistics get applied everywhere, linear algebra has a lot of application.
– Physics: very generally, physics study symmetries of the world. How do they represent symmetries? Through group of linear functions. At the most basic level, everyday life geometry can be represented this way, your symmetries are translation and rotation (this has application in gaming, robotic, aviation, etc.). At the more advanced level, you have gauge theory. And that’s not to mention other application in physics that’s hard to explain in ELI5.
The above didn’t even include matrix not being used in the context of linear algebra, which is just an array of number.
Structural engineers use matrices all the time. In first year statics we’d have to solve 6 equation, 6 variable systems for 3D problems. You’d have equations for strain in all 3 dimensions, and displacement in all 3 dimensions. That gives you a 6×6 matrix which you’d then have to reduce in order to solve the equations.
I would add to posters before me, tons of uses in physics, both classical (Hamiltonian, Lagrangian mechanics), relativistic and quantum.
All kinds of engineering, including electrical, robotics, structural, traffic
Sometimes in cryptography (McEliece/Niederreiter). This one is cool because it stems from a more general application to error correction (which AFAIK is also used to make internet work)
As stated above in statistics (but it get dangerous, don’t confuse with computer Multi-Arrays, which don’t use the row-column product)
They are *everywhere* in math, just way to numerous to even know all the applications. I can scarcely think of a single course that has not involved matrices in one way or another
I’ll add more if I come up with anything
Edit 1: Game theory (a branch of economics, probably more in this field)
Linguistics (Markov chains)
Former game developer here,
Linear algebra are the bread and butter in 2D and 3D video games and photo/image editing/composition software. A single matrix represents the whole universe, matrices represent rotations, translations, and other transformations, they are stored in hierarchies to represent systems of objects and rotations (think of all the joints in a skeleton), they even represent the camera that gives you a perspective view where things in the distance are smaller, and the whole universe is flattened so it can be mapped to your screen.
Matrices can represent systems, from pipes, to circuits, to traffic.
And matrices are used in the original Page Rank algorithm.
That’s just what I can think of off hand.
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