Matrix inequalities and partial isometries that arise in x-ray crystallography Original Research Article

In x-ray crystallography, one needs to consider matrices of the form P = (XTX)/M, where X εimageM × n has rows Xi. with 1 less-than-or-equals, slant i less-than-or-equals, slant M such that ¦Xi.v¦ greater-or-equal, slanted c and short parallelXi.short parallel = 1 for some given c ε (0, 1) and v εimagen. 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 and Kräuter and confirm their conjecture on the subject.