The general common nonnegative-definite and positive-definite solutions to the matrix equations AXA∗ = BB∗ and CXC∗ = DD∗

AbstractWe give necessary and sufficient conditions for the existence of a common nonnegative-definite (positive-definite) solution to the pair of matrix equations AXA∗ = BB∗ and CXC∗ = DD∗, and derive a representation of the general common nonnegative-definite (positive-definite) solution to these two equations when they have such common solutions. This paper can be viewed as a supplementary version of that derived by Young et al. [1] since Groβ [2] has given a counterexample to point out a mistake in their basic Theorem 1. The presented example shows the advantage of the proposed approach

The largest virialized dark halo in the universe

Using semi-analytic approach, we present an estimate of the properties of the largest virialized dark halos in the present universe for three different scenarios of structure formation: SCDM, LCDM and OCDM models. The resulting virial mass and temperature increase from the lowest values of $1.6 \times 10^{15}h^{-1}M_{\odot}$ and 9.8 keV in OCDM, the mid-range values of $9.0 \times 10^{15}h^{-1}M_{\odot}$ and 31 keV in LCDM, to the highest values of $20.9 \times 10^{15}h^{-1}M_{\odot}$, 65 keV in SCDM. As compared with the largest virialized object seen in the universe, the richest clusters of galaxies, we can safely rule out the OCDM model. In addition, the SCDM model is very unlikely because of the unreasonably high virial mass and temperature. Our computation favors the prevailing LCDM model in which superclusters may be marginally regarded as dynamically-virialized systems.Comment: 5 pages, Accepted by Int. J. Mod. Phys.