MATRIX INEQUALITIES AND PARTIAL ISOMETRIES THAT ARISE IN X-RAY CRYSTALLOGRAPHY

In x-ray crystallography, one needs to consider matrices of the form P = (X(T)X)/M, where X is an element of R(M x n) has rows X(i), with 1 less than or equal to i less than or equal to M such that /X(i).v/ gre ater than or equal to c and parallel to x(i). = 1 for some given c is an element of (0, 1) and v is an element of R(n). Using the theory of majorization, we give a short proof for some inequalities relating the eigenvalues of P-1 when P is invertible. Matrices X that minimize det P-1 or tr P-1 are constructed. These extend some results of Ortner an d Krauter and confirm their conjecture on the subject. (C) Elsevier Sc ience Inc., 1997.