XII.—Matrices and Continued Fractions

Abstract

In a communication to this Society in 1916 Professor E. T. Whittaker gave an illuminating exposition of the use of matrices in dealing with continued fractions. In particular he discussed the problem of differentiating the following function of x:—the value of f′(x) being obtained in the first instance from the leading element in the square of a certain reciprocal matrix {M(x)}−1 (cf. formula (9), § 3, below) by a method which I have recapitulated at the opening of § 4. The expression for f′(x) was then given as the quotient of two determinants obtained in a simple and specific manner from the matrix M(x) itself. It was remarked that this treatment of continued fractions takes its rise in a work by Sylvester, who incidentally showed that the fraction f(x) could be expressed as the quotient of two determinants. It was clear that this quotient expressed the leading element in the reciprocal of the matrix M(x): and the question naturally arose, what were the values of the other elements of this reciprocal?