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January 1

Are there any real life applications of matrices?

I want to know the real life applications of matrices. Matrices means finding determinat etc. Rizosome (talk) 23:54, 1 January 2021 (UTC)[reply]

Our article Matrix has a whole section on applications, which is far from exhaustive – for example, it does not even mention any use in the social sciences, such as in principal component analysis. It would be hard to impossible to give a comprehensive overview of all applications, because they are everywhere. They are used in computer graphics such as for computer games and CGI (Computer-generated imagery), in flight control systems, and in GPS. Although determinants are ~~imported~~ important for understanding and developing the theory needed for applications, it is rare that one really needs to compute a determinant. Some much more important computational procedures are Gaussian elimination, matrix decomposition and determining eigenvectors and eigenvalues, which form a basis for many applications. --Lambiam 01:27, 2 January 2021 (UTC)[reply]

Here is a YouTube video on the applications of matrices. --Lambiam 01:41, 2 January 2021 (UTC)[reply]

Is it determinants are imported or important? Rizosome (talk) 14:28, 2 January 2021 (UTC)[reply]

Sorry, typing error; corrected. --Lambiam 15:53, 2 January 2021 (UTC)[reply]

Explained GCI dab above. --CiaPan (talk) 17:10, 2 January 2021 (UTC)[reply]

@Rizosome: Do you consider 'computation' a real life application? If so, please see Kaczmarz method and Algebraic reconstruction technique, both full of linear algebra and both used e.g. in computed tomography. --CiaPan (talk) 17:42, 2 January 2021 (UTC)[reply]

I am asking applications of matrices not about linear algebra. Rizosome (talk) 05:30, 3 January 2021 (UTC)[reply]

Most of modern deep learning is based on matrix multiplication (and other matrix operations). --Stephan Schulz (talk) 18:12, 2 January 2021 (UTC)[reply]

@Rizosome: stochastic matrices come up fairly often in statistical models. Many linear algebra classes will introduce the concept through examples like forecasting weather; while they're not really used in that context very often, the underlying logic is valuable there, and you can readily see application value. (Stochastic matrices are often used in more advanced topics which are hard to introduce to students -- so while weather forecasting is not a true 'real life application', it's close!) Our page on Markov chains has other applications that stochastic matrices may apply to. This is one of my favorite applications of matrices to mention to students. Urve (talk) 21:49, 2 January 2021 (UTC)[reply]

I work extensively with correlation and covariance matrices in the analysis of financial markets and other economic variables. In this context these matrices are used to describe the relationships between changes in different but related financial or other asset prices. In particular, the eigenvectors of these matrices can often give us some understanding of the factors driving these changes. RomanSpa (talk) 23:26, 2 January 2021 (UTC)[reply]

@Rizosome: Saying you're asking about applications of matrices, but not linear algebra, is frankly nonsense. Matrices per se (as opposed to mere arrays of numbers) are always about linear algebra. 100% of the time with no exceptions whatsoever. If I'm wrong about this, saying it this way is the best way to find out :-)

Students are unfortunately typically introduced to the mechanics of manipulating matrices, multiplying them and taking their determinants, before they learn about vector spaces, and then the definitions look silly and arbitrary. But they aren't. A matrix is a way of representing a linear transformation between vector spaces. Always.

Matrix multiplication represents first applying one linear transformation, then the other.

The determinant is a key invariant of a linear transformation, that tells you lots of useful stuff about it. --Trovatore (talk) 07:39, 3 January 2021 (UTC)[reply]

The same is explained in the YouTube video I linked to above in the 1:15–2:48 segment, beginning with "... but the first thing we need to realize is that matrices do things to vectors" and culminating in "These linear transformations are why we call the first in-depth class on matrices 'Linear Algebra'." That said, there are applications of matrices where the elements do not come from a ring but a semiring, such as the tropical semiring. If the partial order of an ordered semigroup gives rise to a lattice, you also get a semiring; see also: Cuninghame-Green, Raymond (1979). Minimax Algebra. Lecture Notes in Economics and Mathematical Systems 166. Berlin–New York: Springer-Verlag. ISBN 978-3-540-09113-4. Matrices are all over the place in this monograph. Calling this "linear algebra" is a bit of a stretch. --Lambiam 08:59, 3 January 2021 (UTC)[reply]

Exactly right, and I'd put it more simply: in the finite-dimensional case, matrices are a completely general way of describing linear transformations: the row space of the matrix describes the part of the source of the linear transformation that is not the kernel and the column space describes the image. This observation is essentially all you need to see the truth of the rank–nullity theorem in the finite-dimensional case. — Charles Stewart (talk) 13:14, 3 January 2021 (UTC)[reply]

@Rizosome: Try solving the following real life problem. I import a certain product and resell it. I can buy the stuff I sell from 3 companies, let's call them A, B, and C. And I can resell what I import via two different stores, let's call them U and V. The contracts I have with A, B, and C allow me to buy a maximum of 700 products per month from A, 900 from B and 800 from C. Shipping costs per product are as follows. From A to U it is $6, from A to V it is $5, from B to U it is $4, from B to V it is $8, from C to U it is $2 and from C to V it is $5.

Customers have already placed orders at the stores U and V for the next month. I have to make sure store U gets supplied with 1000 products, while I need to move 1400 units to store V. I need to figure out how many products from each company are going to be moved to each store such that the total shipping costs are going to be minimized. Count Iblis (talk) 13:48, 3 January 2021 (UTC)[reply]

@Urve: I would love to hear more about that weather forecasting example even though it isn't "real." Sounds fascinating.

I did not see your reply. See Examples of Markov chains#A simple weather model for a good overview of what I mean. The examples I can find outside of Wikipedia are fairly technical and don't do much supporting. The idea is part of Markov chains, as the title of that article suggests. Basically - if we know the weather one day and have a large historical dataset, how can we figure out the probability of the next? Stochastic matrices are one way. Urve (talk) 00:22, 9 January 2021 (UTC)[reply]