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Register a new userRoths solvability criteria for the matrix equations ${AX-widehat XB=C}$ and ${X-Awidehat{X}B=C}$ over the skew field of quaternions with an involutive automorphism $qmapsto hat q$
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The matrix equation $AX-XB=C$ has a solution if and only if the matrices [A&C0&B] and [A &00 & B] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) obtained an analogous criterion for the matrix equation $X-AXB=C$ over a field. We extend these criteria to the matrix equations $AX-widehat XB=C$ and $X-Awidehat XB=C$ over the skew field of quaternions with a fixed involutive automorphism $qmapsto hat q$.
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L.Huang [Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix $A$ with respect to consimilarity transformations $tilde{S}^{-1}AS$ in which $S$ is a nonsingular quaternion matrix and $tilde{h}:=a-bi+cj-dk$ for each quaternion $h=a+bi+cj+dk$. We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations $hat{S}^{-1}AS$ in which $hmapstohat{h}$ is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations $AX-hat{X}B=C$ and $X-Ahat{X}B=C$.
We provide necessary and sufficient conditions for the generalized $star$-Sylvester matrix equation, $AXB + CX^star D = E$, to have exactly one solution for any right-hand side E. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $star$-Sylvester equations.
The treated matrix equation $(1+ae^{-frac{|X|}{b}})X=Y$ in this short note has its origin in a modelling approach to describe the nonlinear time-dependent mechanical behaviour of rubber. We classify the solvability of $(1+ae^{-frac{|X|}{b}})X=Y$ in general normed spaces $(E,|cdot|)$ w.r.t. the parameters $a,binmathbb{R}$, $b eq 0$, and give an algorithm to numerically compute its solutions in $E=mathbb{R}^{mtimes n}$, $m,ninmathbb{N}$, $m,ngeq 2$, equipped with the Frobenius norm.
A theory of monoids in the category of bicomodules of a coalgebra $C$ or $C$-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a $C$-ring (termed a {em matrix $C$-ring}) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a {em Galois module} is introduced and the structure theorem, generalising Schneiders Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167--195], is proven. This is then applied to the $C$-ring associated to a weak entwining structure and a structure theorem for a weak $A$-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or {em infinite matrix ring contexts}) is outlined. A Galois connection associated to a matrix $C$-ring is constructed.
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