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...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/593549 |
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| Summary: | 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. |
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| ISSN: | 1110-757X 1687-0042 |