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Using the ranks and Moore-Penrose inverses of involved matrices, in this paper we establish some necessary and sufficient solvability conditions for a system of Sylvester-type quaternion matrix equations, and give an expression of the general solution to the system when it is solvable. As an application of the system, we consider a special symmetry solution, named the η -Hermitian solution, for a system of quaternion matrix equations. Moreover, we present an algorithm and a numerical example to verify the main results of this paper. [ABSTRACT FROM AUTHOR]

Orchidaceae is the 3rd largest family of angiosperms, an evolved young branch of monocotyledons. This family contains a number of economically-important horticulture and flowering plants. However, the limited availability of genomic information largely hindered the study of molecular evolution and phylogeny of Orchidaceae. In this study, we determined the evolutionary characteristics of whole chloroplast (cp) genomes and the phylogenetic relationships of the family Orchidaceae. We firstly characterized the cp genomes of four orchid species: Cremastra appendiculata, Calanthe davidii, Epipactis mairei, and Platanthera japonica. The size of the chloroplast genome ranged from 153,629 bp (C. davidi) to 160,427 bp (E. mairei). The gene order, GC content, and gene compositions are similar to those of other previously-reported angiosperms. We identified that the genes of ndhC, ndhI, and ndhK were lost in C. appendiculata, in that the ndh I gene was lost in P. japonica and E. mairei. In addition, the four types of repeats (forward, palindromic, reverse, and complement repeats) were examined in orchid species. E. mairei had the highest number of repeats (81), while C. davidii had the lowest number (57). The total number of Simple Sequence Repeats is at least 50 in C. davidii, and, at most, 78 in P. japonica. Interestingly, we identified 16 genes with positive selection sites (the psbH, petD, petL, rpl22, rpl32, rpoC1, rpoC2, rps12, rps15, rps16, accD, ccsA, rbcL, ycf1, ycf2, and ycf4 genes), which might play an important role in the orchid species' adaptation to diverse environments. Additionally, 11 mutational hotspot regions were determined, including five non-coding regions (ndhB intron, ccsA-ndhD, rpl33-rps18, ndhE-ndhG, and ndhF-rpl32) and six coding regions (rps16, ndhC, rpl32, ndhI, ndhK, and ndhF). The phylogenetic analysis based on whole cp genomes showed that C. appendiculata was closely related to C. striata var. vreelandii, while C. davidii and C. triplicate formed a small monophyletic evolutionary clade with a high bootstrap support. In addition, five subfamilies of Orchidaceae, Apostasioideae, Cypripedioideae, Epidendroideae, Orchidoideae, and Vanilloideae, formed a nested evolutionary relationship in the phylogenetic tree. These results provide important insights into the adaptive evolution and phylogeny of Orchidaceae. [ABSTRACT FROM AUTHOR]