Completing a symmetric 2 × 2 block matrix and its inverse Original Research Article

We consider the following completion problems. Suppose n1, n2 are nonnegative integers such that imagen1 +n2 = n> 0 Let A11, A12, A21, B22 be matrices with dimensions n1 × n1, n1 × n2, n2 × n1, and n2 × n2, respectively. We determine necessary and sufficient conditions so that there exists an n2 × n2 matrix A22 such that image and (i) A is nonsingular and symmetric, and B22 is the lower right block of a partitioning of A−1; (ii) A is symmetric positive definite, and B22 is the lower right block of a partitioning of A−1.