Abstract
The T-congruence Sylvester equation is the matrix equation AX+XTB=C, where A∈Rm×n, B∈Rn×m, and C∈Rm×m are given, and X∈Rn×m is to be determined. Recently, Oozawa et al. discovered a transformation that the matrix equation is equivalent to one of the well-studied matrix equations (the Lyapunov equation); however, the condition of the transformation seems to be too limited because matrices A and B are assumed to be square matrices (m=n). In this paper, two transformations are provided for rectangular matrices A and B. One of them is an extension of the result of Oozawa et al. for the case m≥n, and the other is a novel transformation for the case m≤n.
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