Some questions on Matrices.

mickmac76

Hi
I'm just revising leaving cert mathematics from a long time ago and have a question on matrices. The book I'm using defines a matrix as an array of numbers and that's it. I'm trying to figure out what's the point of them. I know how to add, subtract and multiply them, find determinants etc. But what were they invented for if invented is the right word to use. And why the convoluted method of multiplication instead of directly multiplying the matching entries in each matrix. It's just that the whole subject of matrices seems pointless at the moment.

Yakuza

They can be used to solve simultaneous equations, used to find long term positions in stochastic (random/Markov) processes, used to set up differential equations, used in 3D calculations (rotations, translations etc). I think I've only scratched the surface of their power.

JRant

Matrices are a wonderful tool in mathematics. They are used extensively in electrical engineering and quantum theory. propably 2 of the most important subjects in the modern world.

Muppet Man

Matrix manipulation is used extensively in gaming and computer graphics where speed of generating a result is critical.

Michael Collins

Matrix multiplication is defined so that linear transformations can be composed... So while it seems a strange technique, there's a very clear reason.

This is often the case with things that seem strange in maths, such a shame more effort isn't placed on explaining why these seemingly random/weird definitions are defined the way they are...

I'll explain this better when I get a chance, if anyone is interested?

Digital Solitude

I'd like to read that, i do a lot of maths involving matrices and I've no idea what they are outside of vector representation

httpete

If you are building some machinery, such as, say, an aircraft, wouldn't you like to know how the airflow interacts with the geometry of the aircraft? Or how much heats gets generated in different parts of the aircraft during different stages of operation? Or how much vibration is generated and whether the aircraft is structurally unstable?

These problems are formulated as partial differential equations (PDEs) and involve infinite degrees of freedom. Essentially you have an infinite number of equations to solve for an infinite number of unknowns and this is generally impossible to find an analytic solution for, unless the geometry is trivial such as in the case of sphere for example.

So you need to solve them on a computer and to do this you have to transform the infinite dimensional representation into a finite dimensional form. This can be done with schemes such as the Finite Element method or the Boundary Element method in which the true geometry of the aircraft is represented by a mesh of simple shapes/elements like triangles, on each of which the PDEs are required to hold. These approaches result in approximate solutions that converge towards the true solution of the problem if you use more elements in the mesh.

With formulation such as FEM or BEM, or other such as the Nystom method, you ultimately end up with a (very large) finite dimensional system of equations that represents your infinite dimensional problem. This linear system is expressed in the well-known linear algebra matrix vector form as Ax = b, where A is matrix that represents your original equation. Matrices are fundamental to numerically solving PDEs and PDEs represent fundamental physical phenomena such as heat, sound, electromagnetics, fluids, etc.

httpete

Digital Solitude wrote: »

I'd like to read that, i do a lot of maths involving matrices and I've no idea what they are outside of vector representation

They represent linear maps between vector spaces. Suppose you have Ax = b. Each row in the a matrix A is essentially a linear functional that acts on the vector x. A linear functional being an object that takes on an element of the vector space and returns a real number.

Operator theory which deals with linear operators acting on functions can be to an extent viewed as infinite dimensional matrices acting on infinite dimensional vectors. It is really helpful to view functions as infinite dimensional vectors in a lot of situations.