## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2013, Special Issue (2012), Article ID 271978, 5 pages.

### Completing a $2\times2$ Block Matrix of Real Quaternions with a Partial Specified Inverse

#### Abstract

This paper considers a completion problem of a nonsingular $2\times2$ block matrix over the real quaternion algebra $\Bbb H$: Let ${m}_{1},\mathrm{ }{m}_{2},\mathrm{ }{n}_{1},\mathrm{ }{n}_{2}$ be nonnegative integers, ${m}_{1}+{m}_{2}={n}_{1}+{n}_{2}=n>0$, and ${A}_{12}\in {\Bbb H}^{{m}_{1}\times{n}_{2}},\mathrm{ }{A}_{21}\in {\Bbb H}^{{m}_{2}\times{n}_{1}},\mathrm{ }{A}_{22}\in {\Bbb H}^{{m}_{2}\times{n}_{2}},\mathrm{ }{B}_{11}\in {\Bbb H}^{{n}_{1}\times{m}_{1}}$ be given. We determine necessary and sufficient conditions so that there exists a variant block entry matrix ${A}_{11}\in {\Bbb H}^{{m}_{1} \times {n}_{1}}$ such that $A=(\begin{smallmatrix}{A}_{\mathrm{11}}& {A}_{\mathrm{12}}\\[5pt] {A}_{\mathrm{21}}& {A}_{\mathrm{22}}\end{smallmatrix})\in {\Bbb H}^{n \times n}$ 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.

#### Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2012), Article ID 271978, 5 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807327

Digital Object Identifier
doi:10.1155/2013/271978

Mathematical Reviews number (MathSciNet)
MR3045377

#### Citation

Lin, Yong; Wang, Qing-Wen. Completing a $2\times2$ Block Matrix of Real Quaternions with a Partial Specified Inverse. J. Appl. Math. 2013, Special Issue (2012), Article ID 271978, 5 pages. doi:10.1155/2013/271978. https://projecteuclid.org/euclid.jam/1394807327

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