What is a Matrix and why are Matrix operation defined as they are

[CasualKilla]

There seems to be countless resources that describe how to draw a matrix, how to number the elements, how to add matrices, multiply them etc.

 

However, there seems to be very little information describing what exactly the matrix represent and why it is important. Also, why are addition and multiplication defined as they are?

 

Matrices always felt very non-intuitive to me and I hope by achieving better fundamental understanding I can overcome this.

 

If you know of a resource that describes this well or would like to give your own explanation please do.

[imatfaal]

Gilbert Strang's lectures online will give a really good mathematical grounding - as you learn what they do, how to manipulate, etc you start to realise how they represent an integrated and interlinked set of variables.

 

A set of variables will have different (possibly) dimensions - ie the height, hair colour, marital status of a group of forum posters. None of these have a real connexion. Some sets of variables are really just different versions of the same core variable. But matrices elements are seperate but connected

[studiot]

A Matrix isn't anything by itself. It is basically a container that allows us to handle or present all the entries at once.

This is achieved by a formalised structure where position is important.

In that it is like a spreadsheet.

 

A simple device (not a matrix) is the place system in our decimal number system.

 

So what is 9763 ?

 

We are all so familiar with the 'container' that we forget it but it is a way of handling thousands, hundreds, tens and units all at once, if we follow the rules of addition and multiplication.

Note that these rules work well, but [9 7 6 3] is not a matrix and the rules of matrix addition and multiplication do not hold in the place system.

 

So why have different rules for matrices?

 

Well matrix theory is part of linear algebra and the rules for manipulating matrices conforms to the rules of linear algebra.

(Note the elements in a particular position in a matrix may or may not be linear).

 

You have probably encountered other structures that allow us to handle all the entries at once for instance

 

Sequences, summations of series, continued products.

 

Matrices are particularly useful in handling systems of simultaneous equations and in creating data tables for entry into and extraction from computers.

[Prometheus]

The way i like to think about it (which i warn you may not be correct) is in terms of square matrices and the effect they have on a shape, say.

 

Take a 2x2 matrix and a triangle on the Cartesian plane, then applying the matrix to the triangle will do something to the triangle on the plane - maybe its a different kind of triangle or maybe its a line or maybe we just rotated or reflected it. This also helps me make sense of the determinant of the matrix (if i remember correctly!), a determinant of one meaning that whatever we have done to the triangle it has the same area as it did before.

 

That's how i like to think of it anyway, be interesting to know how accurate this is. Doesn't work for non-square matrices though.

[ajb]

Well matrix theory is part of linear algebra and the rules for manipulating matrices conforms to the rules of linear algebra.

 

Indeed, matrices are usually introduced as representations of linear transformations; that is as a concrete way to represent maps from one vector space to another once bases have been chosen.

 

One way to understanding why matrices are multiplied together the way they are is so that composition of linear transformations is represented by matrix multiplication.

[studiot]

Just to stress that whilst matrix algebra is linear the elements need not be.

 

I think Casualkilla is studying something to do with electrical engineering so here is an ee example for discussion, from Morton.

 

Note the matrix admittance equation

 

= [Y] [V]

 

is linear but the matrix elements are decidedly not (eg the resistors in parallel)

 

Edited August 29, 2015 by studiot

[Acme]

There seems to be countless resources that describe how to draw a matrix, how to number the elements, how to add matrices, multiply them etc.

 

However, there seems to be very little information describing what exactly the matrix represent and why it is important. Also, why are addition and multiplication defined as they are?

 

Matrices always felt very non-intuitive to me and I hope by achieving better fundamental understanding I can overcome this.

 

If you know of a resource that describes this well or would like to give your own explanation please do.

This all strikes me as more of an historical question than a mathematical question. In that vein -correct or not- here's some interesting history.

 

The History of the Matrix

The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the first known example of the use of matrix methods to solve simultaneous equations.

 

In the treatise's seventh chapter, "Too much and not enough," the concept of a determinant first appears, nearly two millennia before its supposed invention by the Japanese mathematician Seki Kowa in 1683 or his German contemporary Gottfried Leibnitz (who is also credited with the invention of differential calculus, separately from but simultaneously with Isaac Newton).

 

More uses of matrix-like arrangements of numbers appear in chapter eight, "Methods of rectangular arrays," in which a method is given for solving simultaneous equations using a counting board that is mathematically identical to the modern matrix method of solution outlined by Carl Friedrich Gauss (1777-1855), also known as Gaussian elimination.

I can't find exactly what the simultaneous equations applied to that the Chinese needed to solve, and the link to Nine Chapters of the Mathematical Art is written in German and far beyond my rudimentary 'Wie gehts?'. Anyway, matrices work, i.e. accomplish what is needed to accomplish, and that's why they were and are used. So too the specifics of their use and operations work and that's why they are as they are. Matrices are as matrices do my momma used to say.

 

EDIT: I found a sample matrix problem from Nine Chapters of the Mathematical Art which may illustrate the point I was trying to make, namely that necessity is the mother of invention and sometimes discovery.

 

Rod calculus

Chapter Eight Rectangular Arrays of Jiuzhang suanshu provided an algorithm for solving System of linear equations by method of elimination:[6]

 

Problem 8-1: Suppose we have 3 bundles of top quality cereals, 2 bundles of medium quality cereals, and a bundle of low quality cereal with accumulative weight of 39 dou. We also have 2, 3 and 1 bundles of respective cereals amounting to 34 dou; we also have 1,2 and 3 bundles of respective cereals, totaling 26 dou.

 

Find the quantity of top, medium, and poor quality cereals. In algebra, this problem can be expressed in three system equations with three unknowns.

Edited August 29, 2015 by Acme

[Sriman Dutta]

Matrix is a way of handling and using multiple numbers at the same time simultaneously by having definite matrix operation rules.

 

You may think a matrix like an array in OOP language (though it's not totally correct to think it in that way).