What Are the Various Applications of Matrices Beyond Solving Equations?

[DeadWolfe]

I've only taken one linear algebra course, and when I was tought about matricies, I immediatly wondered what the point of them was. I understand they can be useful for solving systems of equations, but there must be other things they can be used for. Probably many.

...what are these things?

 

[Zurtex]

I think you belittle solving systems of equations, this is endlessly useful particularly in numerically approximating solutions to differential equations.

Right now I am using properties of matrices to help me with a problem about cyclic permutations, as I know and can manipulate matrices much easier and it’s really easy to come up with equivalent mtracies.

 

[masudr]

Representations of groups? Representation of linear operators in terms of particular bases? Representing observables in QM? The list goes on...

 

[robphy]

Google [graph theory]
http://www.mathworks.com/company/newsletters/news_notes/clevescorner/oct02_cleve.html

 

[Genecks]

Creating a virtual prison for the mind. Wouldn't love to be able to create a virtual reality world which you could tap into and perhaps access the internet from? TONS OF THINGS! Math is used for loads of things! Artificial intelligence and etc etc.

 

[Diophantus]

I would just like to challenge the view that maths has to have 'uses'. I like to study maths purely because I derive much enjoyment from doing so.

You can find many enjoyable results in linear algebra without having to view them in terms of their applications and I think linear algebra is an excellent setting to begin to understand tecniques of proof.

Incidentally, in my opinion applications of mathematics are just as likely to be used to the detriment of mankind as they are for the benefit. Two examples of the potential detrimental effects of applied maths to mankind are atomic weaponry (and, in fact, warfare in general) and applications which advance a people economically which, one could argue, can cause great disparity in wealth distribution.

I am not arguing one way or the other as to the effect of applied maths (defining that term is also a topic worthy of much debate) on mankind, I am merely challenging a common belief that maths should be judged on utilitarian grounds.

I hope I do not come across as a pedant, I am merely trying to highlight a very subtle point. I know that this is not exactly what you were getting at but having recently read Hardy's A Mathematician's Apology, I felt an urge to make share my new views.

 

[mathwonk]

i like to quote the great emil artin who said essentially matrices should be left out of linear algebra whenever possible. They are sometimes needed for computing a determinant and that's about it.

they should be introduced at that point, the determinant calculated, and then they should be thrown out again.

 

[slayerchange]

I've only taken one linear algebra course, and when I was tought about matricies, I immediatly wondered what the point of them was. I understand they can be useful for solving systems of equations, but there must be other things they can be used for. Probably many.
...what are these things?

One general and natural technique in mathematics is to approximate any system via linear systems. These linear systems are especially vector spaces or normed spaces.
Matrices are used for repesenting linear transformations between these spaces. I.e, A matrix is a function between two linear space. Here is some examples:
-solving linear systems
-reperesenting linear transformations: like jacobian matrix,mobius transformations(groups),symetry groups( rotation matrix),fundamental matrices( for differential equations), or matrices for independency(diff. equations,algebraic number theory) , ..
-Evaluating volume of higher dimensional objects
-Sometimes, like Lorentz transformations, they can be used for physical applications as a reperesentitive of certain transformations.
-they are also used for difference equations
-in differential geometry,they are used for expansions of Kristofell symbols..