Completing a symmetric \(2\times 2\) block matrix and its inverse
The author considers the following completion problem: suppose we are given two nonegative integers \(n_1\), \(n_2\) such that \(n_1+ n_2= n>0\). Also suppose that we are given the matrices \(A_{11}\), \(A_{12}\), \(A_{21}\), and \(B_{22}\) having dimensions \(n_1 \times n\), \(n_1 \times n_2\), \(n_2 \times n_1\), and \(n_2 \times n_2\), respectively. Then the author determines necessary and sufficient conditions so that there exists an \(n_2 \times n_2\) matrix \(A_{22}\) such that \[ A= \left[ \begin{matrix} A_{11} &A_{12}\\ A_{21} &A_{22} \end{matrix} \right] \] and i) \(A\) is nonsingular and symmetric, and \(B_{22}\) is the lower right block of a partitioning of \(A^{-1}\), ii) \(A\) is symmetric positive definite, and \(B_{22}\) is the lower right block of a partitioning of \(A^{-1}\).
scientific article; zbMATH DE number 871123
scientific article