On the Hilbert matrix
Tosio Kato · Proceedings of the American Mathematical Society · 1957
The present paper is concerned with the existence2 of the eigenvalue ir of the Hilbert matrix A =((i+k -1)_1) or A = ((i+k)'1), i, k = l, 2, 3, • • • .It is well known that,3 considered as a linear operator in the Hilbert space I2 of vectors with finite square sum of components, A is symmetric, positive-definite and bounded, the upper bound being equal to ir.It is further known that4 ir is not an eigenvalue of A thus defined.However, the question has remained open whether there exists any eigenvector (not belonging to the Hilbert space) with the eigenvalue ir of the matrix A.In what follows we shall show that there exists such an eigenvector x and that x may be chosen positive.Further we shall show that x is logarithmically convex in the sense that [x(i-\-l)]2^x(i) ■x(i+2).Actually we shall establish these results for a rather wide class of matrices containing the Hilbert matrix as a special case.Our method is quite simple and elementary:we consider the dominant eigen-vectors6 of the nXn segments A" of A and show that the ith components of these eigenvectors form (when properly normalized), for each fixed i, a monotone converging sequence; the limiting vector thus obtained being shown to be the required eigenvector of A.These results may be of some interest in view of various numerical work6 done recently on the segments An of the Hilbert matrix.Actually the present investigation was suggested by a table7 of the dominant eigenvectors of An.1.In what follows we consider matrices A = (a(i, k)) which may be finite or infinite, square or rectangular.In any case we assume that the indices i, k take on positive integral values starting with 1, that is, i=l, 2, • • ■ , m(A) and k = l, 2, ■ ■ ■ , n(A), where m(A) and n(A), which may be finite or infinite, denote respectively the number of