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:
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 by Acme, 29 August 2015 - 09:18 PM.