Completing a 2 × 2 Block Matrix of Real Quaternions with a Partial Specified Inverse

Abstract

This paper considers a completion problem of a nonsingular $2\times 2$ block matrix over the real quaternion algebra $ℍ$: Let $m_1,m_2,n_1,n_2$ be nonnegative integers, $m_1+m_2=n_1+n_2=n>0$, and $A_{12}\in {ℍ}^{m_1\times n_2},\mathrm{ }{A}_{21}\in {ℍ}^{m_2\times n_1},\mathrm{ }{A}_{22}\in {ℍ}^{m_2\times n_2},\mathrm{ }{B}_{11}\in {ℍ}^{n_1\times m_1}$ be given. We determine necessary and sufficient conditions so that there exists a variant block entry matrix $A_{11}\in {ℍ}^{m_1\times n_1}$ such that is nonsingular, and $B_{11}$ is the upper left block of a partitioning of $A^{-1}$. The general expression for $A_{11}$ is also obtained. Finally, a numerical example is presented to verify the theoretical findings.