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An Introduction To Non-aristotelian Systems And General Semantics.

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THE NEWER 'MATTER'
707
The notation by suffixes is very convenient and is very much used these days. The first suffix denotes the row, the second the column in which the element is situated. Usually the comma dividing the two numbers in the index is dropped and the coefficients are written simply an, instead of ai,\. In general the element ttj,*, or a,*, represents the element in row i and column k.
The elements lying in the diagonal joining the upper left-hand to the lower right-hand number are called the principal diagonal. In our example we notice that the elements in the diagonal are such that ik.
We have definite rules by which we can arrive at the solution of our equations, once the coefficients, which are the elements of the determinant, are given. In general, the determinants are treated as a functional form.
If m and n are positive integers, a manifold, or system of mn ordered quantities or elements arranged in m horizontal and n vertical rows, will be called a rectangular matrix and we may use the notation:
The numbers m and « are called the orders of the matrix. If m =« the matrix is called a square matrix. Without loss of generality we can treat any rectangular matrix in whichas a square matrix by supplementing the missing rows
and columns with zeros.
A matrix of the type,
where for, is called a unit matrix.
The matrix
is called a diagonal matrix. In the new quantum mechanics a diagonal matrix is independent of t and represents a constant of the classical theory. The reverse is not necessarily true. The operation of differentiation can be expressed in terms of multiplication of matrices with the aid of the unit matrix.4
Equations in which matrices are equated are called matrix equations. If the equations involve only one unknown matrix, which does not occur more than once as a factor, such equations are called matrix equations of the first degree.