Ranks of Solutions of the Linear Matrix Equation AX + YB = C

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Abstract

For a consistent complex matrix equation AX + YB = C, we solve the following two problems:

  • (1)

    the maximal and minimal ranks of a pair of solutions X and Y to AX + YB = C, and

  • (2)

    the maximal and minimal ranks of four real matrices X0, X1, Y0, and Y1 in a pair of solutions X = X0 + iX1 and Y = Y0 + iY1 to AX + YB = C.

We also give a necessary and sufficient condition for matrix equations AiXi + YiBi = C (i = 1, 2) to have common solutions.

Keywords

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

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