NUMERICAL ALGEBRA, doi:10.3934/naco.2020016 CONTROL AND OPTIMIZATION MAXIMUM AND MINIMUM RANKS AND INERTIAS OF THE HERMITIAN PARTS OF THE LEAST RANK SOLUTION OF THE MATRIX EQUATION AXB=C Sihem Guerarra * Faculty of Exact Sciences and Sciences of Nature and Life Department of Mathematics and informatics University of Oum El Bouaghi, 04000, Algeria (Communicated by Yongzhong Song) Abstract. In this paper we derive the extremal ranks and inertias of the matrix X + X * − P , with respect to X, where P ∈ C n×n H is given, X is a least rank solution to the matrix equation AXB = C, and then give necessary and sufficient conditions for X + X * ≻ P (≥ P , ≺ P , ≤ P ) in the L¨ owner partial ordering. As consequence, we establish necessary and sufficient conditions for the matrix equation AXB = C to have a Hermitian Re-positive or Re-negative definite solution. 1. Introduction. Throughout this work, all m × n complex matrices and all n × n complex Hermitian matrices are denoted by C m×n and C n×n H respectively, the symbols, A * , r (A) and I m stand for the conjugate transpose of A, the rank of A and the identity matrix of order m respectively. The Moore-Penrose inverse of a matrix A ∈ C m×n , denoted by A + , is the unique matrix X ∈ C n×m which satisfies the following four matrix equations: (1) AXA = A, (2) XAX = X, (3) (AX) * = AX, (4) (XA) * = XA. The Moore-Penrose generalized inverse has been the subject of many researches, see for example, [1], [2], [9], [10]. We denote E A = I − AA + , F A = I − A + A the two orthogonal projectors induced by A ∈ C m×n such that r (E A )= m − r (A), r (F A )= n − r (A). The inertia of A ∈ C n×n H is the set In (A)= {i + (A), i - (A), i 0 (A)}, where i + (A), i - (A) and i 0 (A) are the numbers of positive, negative and zero eigenvalues of A counted with multiplicities respectively. For a matrix A ∈ C n×n H , we know that r (A)= i + (A)+ i - (A) and i 0 (A)= n − r (A). Consider the linear matrix equation AXB = C , (1) where, A ∈ C m×n ,B ∈ C n×q ,C ∈ C m×q are given, and X ∈ C n×n is unknown. Many cases have been studied about the equation (1), for example, in [17] the author gave necessary and sufficient conditions for the existence of Hermitian nonnegative- definite or positive-definite solutions to (1), then a representation of these solutions 2010 Mathematics Subject Classification. Primary: 15A24; Secondary: 15A09, 15A03, 15B57. Key words and phrases. Matrix equation, Moore-Penrose generalized inverse, Rank, Inertias, Least-rank solution. * Corresponding author: Sihem Guerarra. 1