Solving Optimization Problems on Hermitian Matrix Functions with Applications
We consider the extremal inertias and ranks of the matrix expressions f(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, where A3=A3*, B3, C3, and D3 are known matrices and Y and X are the solutions to the matrix equations A1Y=C1, YB1=D1, and A2X=C2, respectively. As applications, we present necessary and suf...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2013-01-01
|
| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/593549 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832551824826040320 |
|---|---|
| author | Xiang Zhang Shu-Wen Xiang |
| author_facet | Xiang Zhang Shu-Wen Xiang |
| author_sort | Xiang Zhang |
| collection | DOAJ |
| description | We consider the extremal inertias and ranks of the matrix expressions f(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, where A3=A3*, B3, C3, and D3 are known matrices and Y and X are the solutions to the matrix equations A1Y=C1, YB1=D1, and A2X=C2, respectively. As applications, we present necessary and sufficient condition for the previous matrix function f(X, Y) to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations A1Y=C1, YB1=D1, A2X=C2, and B3X+(B3X)*+C3YD3+(C3YD3)*=A3, and give an expression of the general solution to the above-mentioned system when it is solvable. |
| format | Article |
| id | doaj-art-a703efbec34e4fe0a2f4e43b405d4c08 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-a703efbec34e4fe0a2f4e43b405d4c082025-02-03T06:00:31ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/593549593549Solving Optimization Problems on Hermitian Matrix Functions with ApplicationsXiang Zhang0Shu-Wen Xiang1Department of Computer Science and Information, Guizhou University, Guiyang 550025, ChinaDepartment of Computer Science and Information, Guizhou University, Guiyang 550025, ChinaWe consider the extremal inertias and ranks of the matrix expressions f(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, where A3=A3*, B3, C3, and D3 are known matrices and Y and X are the solutions to the matrix equations A1Y=C1, YB1=D1, and A2X=C2, respectively. As applications, we present necessary and sufficient condition for the previous matrix function f(X, Y) to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations A1Y=C1, YB1=D1, A2X=C2, and B3X+(B3X)*+C3YD3+(C3YD3)*=A3, and give an expression of the general solution to the above-mentioned system when it is solvable.http://dx.doi.org/10.1155/2013/593549 |
| spellingShingle | Xiang Zhang Shu-Wen Xiang Solving Optimization Problems on Hermitian Matrix Functions with Applications Journal of Applied Mathematics |
| title | Solving Optimization Problems on Hermitian Matrix Functions with Applications |
| title_full | Solving Optimization Problems on Hermitian Matrix Functions with Applications |
| title_fullStr | Solving Optimization Problems on Hermitian Matrix Functions with Applications |
| title_full_unstemmed | Solving Optimization Problems on Hermitian Matrix Functions with Applications |
| title_short | Solving Optimization Problems on Hermitian Matrix Functions with Applications |
| title_sort | solving optimization problems on hermitian matrix functions with applications |
| url | http://dx.doi.org/10.1155/2013/593549 |
| work_keys_str_mv | AT xiangzhang solvingoptimizationproblemsonhermitianmatrixfunctionswithapplications AT shuwenxiang solvingoptimizationproblemsonhermitianmatrixfunctionswithapplications |