An inverse eigenproblem for generalized reflexive matrices with normal k+1k+1-potencies


Let P,Β Q∈CnΓ—nP,~Q\in\mathbb{C}^{n\times n} be two normal {k+1}\{k+1\}-potent matrices, i.e., PPβˆ—=Pβˆ—P,Β Pk+1=PPP^{*}=P^{*}P,~P^{k+1}=P, QQβˆ—=Qβˆ—Q,Β Qk+1=QQQ^{*}=Q^{*}Q,~Q^{k+1}=Q, k∈Nk\in\mathbb{N}. A matrix A∈CnΓ—nA\in\mathbb{C}^{n\times n} is referred to as generalized reflexive with two normal {k+1}\{k+1\}-potent matrices PP and QQ if and only if A=PAQA=PAQ. The set of all nΓ—nn\times n generalized reflexive matrices which rely on the matrices PP and QQ is denoted by GRnΓ—n(P,Q)\mathcal{GR}^{n\times n}(P,Q). The left and right inverse eigenproblem of such matrices ask from us to find a matrix A∈GRnΓ—n(P,Q)A\in\mathcal{GR}^{n\times n}(P,Q) containing a given part of left and right eigenvalues and corresponding left and right eigenvectors. In this paper, first necessary and sufficient conditions such that the problem is solvable are obtained. A general representation of the solution is presented. Then an expression of the solution for the optimal Frobenius norm approximation problem is exploited. A stability analysis of the optimal approximate solution, which has scarcely been considered in existing literature, is also developed

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University of Wyoming

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