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

Abstract EN

This paper considers a completion problem of a nonsingular 2×2 block matrix over the real quaternion algebra ℍ: Let m1, m2, n1, n2 be nonnegative integers, m1+m2=n1+n2=n>0, and A12∈ℍm1×n2, A21∈ℍm2×n1, A22∈ℍm2×n2, B11∈ℍn1×m1 be given.

We determine necessary and sufficient conditions so that there exists a variant block entry matrix A11∈ℍm1×n1 such that A=(A11A12A21A22)∈ℍn×n is nonsingular, and B11 is the upper left block of a partitioning of A-1.

The general expression for A11 is also obtained.

Finally, a numerical example is presented to verify the theoretical findings.