Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations

Khatri, C. G.; Mitra, Sujit Kumar

doi:10.1137/0131050

Cited by 239 publications

(115 citation statements)

References 7 publications

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“…Let K be a solution of the matrix inequalities (68). Proposition 3 implies that K B E = C T E. Then, [42,Thm. 2.2] implies that the conditions 2b and 2c hold and K must be of the form (69) where X = E T C, Y = E T B T .…”

Section: The Following Conditions Hold: (A) D Is Positive Semi-definimentioning

confidence: 99%

“…The following lemma on positive semi-definite solutions of matrix equations, taken partly from [42], will be needed in the proof of the theorem below. …”

Section: Remarkmentioning

confidence: 99%

“…The generalized inverse (XY T ) − can be taken to be symmetric and positive semidefinite as noted in [42], which entails ker X T (XY T ) − X = ker(XY T ) − X . Moreover, since rank XY T = rank X , any element of the column span of X can be written in the form (70) is shown.…”

Section: Lemma 1 If the Equation Y K = X Where Y And X Are Given Mamentioning

confidence: 99%

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Linear passive systems and maximal monotone mappings

Camlibel

Schumacher

2015

Math. Program.

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This paper deals with a class of dynamical systems obtained from interconnecting linear systems with static set-valued relations. We first show that such an interconnection can be described by a differential inclusions with a maximal monotone set-valued mappings when the underlying linear system is passive and the static relation is maximal monotone. Based on the classical results on such differential inclusions, we conclude that such interconnections are well-posed in the sense of existence and uniqueness of solutions. Finally, we investigate conditions which guarantee well-posedness but are weaker than passivity. Mathematics Subject Classification

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Section: The Following Conditions Hold: (A) D Is Positive Semi-definimentioning

confidence: 99%

“…The following lemma on positive semi-definite solutions of matrix equations, taken partly from [42], will be needed in the proof of the theorem below. …”

Section: Remarkmentioning

confidence: 99%

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Linear passive systems and maximal monotone mappings

Camlibel

Schumacher

2015

Math. Program.

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“…Hermitian positive semidefinite solutions to some matrix equations or some operator equations were investigated by many authors. For finite matrices, Khatri and Mitra [12] For any A ∈ B(H, K), the Moore-Penrose inverse A † of A exists if and only if A has closed range. In this case, A † exists uniquely and (A * ) † = (A † ) * .…”

mentioning

confidence: 99%

Positive and re-positive solutions to some systems of adjointable operator equations over Hilbert C*-modules

Song¹,

Wang²

2011

ELA

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Abstract. A necessary and sufficient condition for the existence of the general common positive solution to equationsfor operators between Hilbert C * -modules is established, and an expression for the common positive solution to the equations is derived when the solvability conditions are satisfied. As an application, a new necessary and sufficient condition for the system of adjointable operator equationsover Hilbert C * -modules to have a common Re-positive solution is proved. Moreover, an expression of the general Re-positive solution is derived when the consistent conditions are met. The results of this paper extend some known results in the literature.

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“…The two-sided matrix equations AXA * = B and AXB = C have many essential applications in mathematics and other fields, and were extensively studied in the literature; see, e.g., [1,3,5,6,7,9,11,13,14,18,19,20]. Concerning the consistency and general solutions of AXA * = B and AXB = C, the following results are well known; see, e.g., [5,7,13,14].…”

Section: Lemma 11 ([14]mentioning

confidence: 99%

On relations among solutions of the Hermitian matrix equation $AXA^{*} = B$ and its three small equations

Li¹,

Tian²

2014

Ann. Funct. Anal.

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Abstract. Assume that the linear matrix equation AXA * = B = B * has a Hermitian solution and is partitioned as. We study in this paper relations among the Hermitian solutions of the equation and the three small-size matrix equations A 1 X 1 A * 1 = B 11 , A 1 X 2 A * 2 = B 12 and A 2 X 3 A * 2 = B 22 . In particular, we establish closed-form formulas for calculating the maximal and minimal ranks and inertias of X − X 1 − X 2 − X * 2 − X 3 , and use the formulas to derive necessary and sufficient conditions for the Hermitian matrix equality X = X 1 + X 2 + X * 2 + X 3 to hold and Hermitian matrix inequalities X > ( , <, ) X 1 + X 2 + X * 2 + X 3 to hold in the Löwner partial ordering.

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Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations

Cited by 239 publications

References 7 publications

Linear passive systems and maximal monotone mappings

Linear passive systems and maximal monotone mappings

Positive and re-positive solutions to some systems of adjointable operator equations over Hilbert C*-modules

On relations among solutions of the Hermitian matrix equation $AXA^{*} = B$ and its three small equations

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