Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices Original Research Article

Let image and image be nontrivial involutions; thus R=R−1≠±I and S=S−1≠±I. We say that image is (R,S)-symmetric ((R,S)-skew symmetric) if RAS=A (RAS=−A). Let image be the class of m×n (R,S)-symmetric matrices or the class of m×n (R,S)-skew symmetric matrices. Let image and image. We study the following problems:(i) Give necessary and sufficient conditions for the existence of an image such that AZ=W, and find all such matrices if the conditions are met.(ii) Find image and characterize the class image.(iii) If image is arbitrary, find image and find image such that short parallelA−Bshort parallel=σ(Z,W,B). We obtain explicit formulas for σ(Z,W), σ(B,Z,W), and all the matrices in question.