Ranks of least squares solutions of the matrix equation AXB=C

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Abstract

For a complex matrix equation AXB=C, we solve the following two problems: (1) the maximal and minimal ranks of least square solution X to AXB=C, and (2) the maximal and minimal ranks of two real matrices X0 and X1 in least square solution X=X0+iX1 to AXB=C. We also give a necessary and sufficient condition for matrix equations AiXiBi=Ci(i=1,2) to have a common least square solution.

Keywords

Matrix equation
Solvability condition
Least square solution
Maximal rank
Minimal rank
Common solution
Generalized inverse
Matrix rank method

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This work was supported by the Foundation of Shanghai Education Committee (07zz171).