To solve matrix equation \(\sum A^iXB_i=C\) by the Smith normal form
Let \(F\) be a field. The paper studies the matrix equation (*) \(\sum _{i=0}^kA^iXB_i=C\) with given matrices \(A\in F^{m\times m}\), \(B_i\in F^{n\times q}\), \(C\in F^{m\times q}\) and unknown matrix \(X\in F^{m\times n}\). The universal solvability of \((*)\) is discussed (i.e. solvability for any \(C\)) and explicit formulas for the general or for the unique solution are given. The proofs use the Smith normal form of a rectangular matrix \(B(\lambda)\in F^{n\times q}[\lambda ]\): there exist unimodular matrices \(P(\lambda)\in F^{n\times n}[\lambda ]\) and \(Q(\lambda)\in F^{q\times q}[\lambda ]\) such that \(P(\lambda)B(\lambda)Q(\lambda)= \left( \begin{smallmatrix} D&0\cr 0&0 \end{smallmatrix}\right) \) where \(D=\) diag\((d_1(\lambda),\ldots ,d_r(\lambda))\), \(d_i\) are scalar monic polynomials (the invariant polynomials of \(B(\lambda)\)) and \(d_i\) is divisible by \(d_{i-1}\). The formulas presenting the solution use the g-inverse of a matrix \(M\in F^{m\times n}\), i.e. a matrix \(M^-\) such that \(MM^-M=M\).
scientific article; zbMATH DE number 1775105