Volterra Multipliers II

Abstract

If p is a real m by m matrix, a Volterra multiplier is a positive diagonal matrix a such that, in this sense of quadratic forms, $ap\leqq 0$. The usefulness of this condition has been well-known since it was introduced by Volterra around 1930, but the usefulness is diminished by the difficulty of deciding whether the multiplier exists. In 1978 the case $m = 3$ was fully solved by Cross, but the case $m\geqq 4$ has remained open except under simplifying assumptions; e.g., that the matrix p is in some suitable graph-theoretic sense sparse. Results under such assumptions were given in Part I (SIAM J. Alg. Disc. Meth., 6 (1985), pp. 570–589. of this study, m being arbitrary. Here we present a general theorem, without supplementary hypotheses, which reduces the problem for m to two simultaneous problems for $m - 1$, in the spirit of the 1978 work cited above. As a consequence we are able to give a complete theoretical solution when $m = 4$ and an effective computational procedure for larger values. In this sense, the Volterra problem can be regarded as solved. The procedure is illustrated by examples which are accessible to previous methods only with difficulty, if at all.