We generalize the theory of positive diagonal scalings of real positive definite matrices to complex diagonal scalings of complex positive definite matrices. A matrix A is a diagonal scaling of a positive definite matrix M if there exists an invertible complex diagonal matrix D such that A = D*MD and where every row and every column of A sums to one. We look at some of the key properties of complex diagonal scalings and we conjecture that every n by n positive definite matrix has at most 2n−1 scalings and prove this conjecture for certain special classes of matrices.We also use the theory of complex diagonal matrix scalings to formulate a van der Waerden type question on the permanent function; we show that the solution of this question would have applications to finding certain maximally entangled quantum states.
@article{bwmeta1.element.doi-10_2478_spma-2014-0007,
author = {Rajesh Pereira and Joanna Boneng},
title = {The theory and applications of complex matrix scalings},
journal = {Special Matrices},
volume = {2},
year = {2014},
pages = {68-77},
zbl = {1291.15080},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0007}
}
Rajesh Pereira; Joanna Boneng. The theory and applications of complex matrix scalings. Special Matrices, Tome 2 (2014) pp. 68-77. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0007/
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