1 5G 
PROF. H. F. BAKER ON CERTAIN LINEAR 
The n 2 quantities can be arranged to form a square of n rows and n columns, 
the first suffix i denoting the row, and the second suffix j denoting the column in 
which a particular element u i} is placed. This square is denoted by a single symbol, 
say u, and called a matrix. The symbol uv, formed from the two symbols u, v, 
written in a definite order, denotes then another matrix whose (i, j) th element has 
the value 
U il V lj + U i2 V 2j + • • • + U in V n j, 
which is formed from the elements of the i th row of the matrix u and those of the 
j th column of the matrix v. This new matrix uv is called the product of u and v, 
taken in this order ; it is generally different from vu. The symbol 1, when used for 
a matrix of an assigned number of rows and columns, denotes the matrix of which 
every element is zero except those in the diagonal, all of which have the same value, 
unity; it is easy to see that any matrix is unaltered by multiplication with the 
matrix unity of the same number of rows and columns. The symbol u~ l denotes 
the matrix such that the product u~ 1 u is the matrix unity; in that case uur 1 is equal 
to u~ l u ; the symbol u~ x is nugatory when the determinant formed with the elements 
of u is zero, and only then. In general, the determinant formed with the elements 
of u will be denoted by \u\. By the sum, u + v, of two matrices u, v, of the same 
number of rows and columns, is meant the matrix whose (i, j) th element is u^+Vy, 
and, similarly, for the difference. Frequently we denote the aggregate of a row of 
n quantities, x l} x 2 , ..., x n by the single letter x ; then if u be a matrix of n rows and 
columns, the symbol ux denotes a set of n quantities of which the i th is 
u il x 1 + u i2 x 2 + ... + u in x n . 
By the differential coefficient of a matrix we mean the single matrix whose elements 
are the differential coefficients of the given one. In what follows, if the ( i , ^) th 
element of a matrix u be a function of t, we denote by Q u the matrix of which the 
(i, j) th element is the integral of u tj taken in regard to t from t — 0 to t = t. If, for 
an instant this matrix Q u be denoted by v, the product matrix uv will be denoted 
by uQu, and the matrix Q (uv), or Q(uQu), will be denoted by Q uQu. Similarly, 
Q ( u . QuQu) will be denoted by QuQuQu, and so on. 
Now consider a matrix of which the (i, j) th element is the infinite series formed by 
the sum of the (i, j ) th elements taken from the matrix unity (of the same number of 
rows and columns as u), the matrix Qu, the matrix QuQu, the matrix QuQuQu, and 
so on. This will be denoted by 
Q (u) = 1 + Qu + QuQu + QuQuQu +..., 
and the series on the right will be said to be uniformly and absolutely convergent 
when this property is proved to hold for each of the n 2 infinite series which constitute 
its elements. 
