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Completing a block diagonal matrix with a partially prescribed inverse. (English) Zbl 0831.15010

A problem of completing a matrix and its inverse is considered: given a field \(F\) and the matrices \(A,B,C\) and \(D\) over \(F\) of sizes \(n\times m\), \(p\times q\), \(m\times p\), and \(q\times n\), respectively, where \(n+p=m+q\), matrices \(W,X,Y\) and \(Z\) of appropriate dimensions have to be found such that \[ \left(\begin{matrix} A & W\\ X & B\end{matrix}\right)^{-1}=\left(\begin{matrix} Y & C\\ D & Z\end{matrix}\right). \] Necessary and sufficient conditions for the existence of a solution are given which in the general case requires the solution of a quadratic matrix equation.
Several special cases are emphasized, which give rise to more easily verified necessary and sufficient conditions.

MSC:

15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
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