THE NEWER 'MATTER' |
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Mathematics recognizes that this generalization of
mathematical notions is extremely useful and legitimate. This structural issue appears to be of very general application, as all of us exhibit a tendency towards it. It is a purely mathematical and useful tendency in mathematics, but it leads to disastrous results when applied to daily-h-i abstractions, as explained in Part VII. In this connection we should recall the difference between the mathematical con-tentless abstractions and the abstractions with physical content, with which we are generally concerned in science and life.Let us now follow up the method by which a matrix can be considered as a quantity. If we have objects of two or more kinds which can be counted or measured, and if we consider an aggregate of such objects, say 5 horses, 3 cows and 2 sheep, we could denote such a complex quantity by the symbol (5,3,2). In this case, the first place in our symbol would be reserved for horses, the second for cows, and the third for sheep.
In mathematics, we do not specify horses, or cows, or sheep, but consider sets of quantities, and distinguish them by the position which they have in our symbolism. We may denote such a complex quantity by a single letter,
A =(a,6,c,). (For instance, we denote a fraction by a single letter, although a fraction is specified by two numbers.)In such an instance, we should call a complex quantity equal to another when, and only when, the components are respectively equal. And a complex quantity is said to vanish only in case all the components vanish.
Ordinary mathematical operations can be applied to such complex quantities. For instance, we may define a sum or difference of two complex quantities
a definition which is entirely satisfactory theoretically, and also practically, as can be verified from our example.
From this point of view we may consider a matrix as a complex quantity with
mn or mcomponents. A matrix would then represent a complex quantity, as a special case under the general method sketched above.^{2} We could then define our further operations. A matrix would be said to be zero when all elements are equal to zero. Two matrices would be said to be equal when they have equal numbers of rows and columns and every element of one is equal to the corresponding element of the other.
By setting up some such rules we could develop a calculus of matrices, and matrices would be considered as complex numbers. In general, the algebraic rules would be found to be applicable to matrices, which would further justify us in treating matrices as complex numbers.
One of the notable exceptions in our operations would be found in the application of the classical operation of multiplication and its dependencies. In ordinary algebra and arithmetic, multiplication is what is called 'commutative' which means that
In defining the multiplication of matrices we have no a priori grounds for determining why one definition or restriction should be preferable to another. |
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