Solutions to optimization problems on ranks and inertias of a matrix function with applications

He, Zhuo-Heng; Wang, Qing-Wen

doi:10.1016/j.amc.2012.09.024

Cited by 11 publications

(8 citation statements)

References 42 publications

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“…By Lemma 1, we obtain that system (80) is solvable if and only if (14), (81) is consistent if and only if (17). The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into 82…”

Section: (78)mentioning

confidence: 95%

“…(g) has a matrix ≥ 0( ≤ 0) if and only if min ∈ − ( ) = 0 (min ∈ + ( ) = 0); (h) any ∈ satisfies ≥ 0 ( ≤ 0) if and only if max ∈ − ( ) = 0 (max ∈ + ( ) = 0). Lemma 6 (see [16]). Let ( , ) = − − ( ) * − − ( ) * , where , , , and are given with appropriate sizes, and denote that…”

Section: Extremal Ranks and Inertias Of Hermitianmentioning

confidence: 99%

“…The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into 82…”

Section: (78)mentioning

confidence: 99%

“…One of his works [15] considered the extremal ranks and inertias of (7). He and Wang [16] derived the extremal ranks and inertias of (7) subject to 1 = 1 , 2 2 = 2 . Liu and Tian [17] studied the extremal ranks and inertias of (8).…”

Section: Introductionmentioning

confidence: 99%

“…respectively. He and Wang [16] derived the extremal ranks and inertias of (10). We consider the extremal ranks and inertias of (10) subject to (11) and…”

Section: Introductionmentioning

confidence: 99%

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Solving Optimization Problems on Hermitian Matrix Functions with Applications

Zhang¹,

Xiang²

2013

Journal of Applied Mathematics

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We consider the extremal inertias and ranks of the matrix expressionsf(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, whereA3=A3*, B3, C3, andD3are known matrices andYandXare the solutions to the matrix equationsA1Y=C1,YB1=D1, andA2X=C2, respectively. As applications, we present necessary and sufficient condition for the previous matrix functionf(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 equationsA1Y=C1,YB1=D1,A2X=C2, andB3X+(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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Section: (78)mentioning

confidence: 95%

Section: Extremal Ranks and Inertias Of Hermitianmentioning

confidence: 99%

“…The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into 82…”

Section: (78)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…respectively. He and Wang [16] derived the extremal ranks and inertias of (10). We consider the extremal ranks and inertias of (10) subject to (11) and…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations