Nonnegative-definite and positive-definite solutions to the matrix equationAXA∗=B – revisited

Groß, Jürgen

doi:10.1016/s0024-3795(00)00033-1

Cited by 58 publications

(26 citation statements)

References 4 publications

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“…In 2000, Groß in [9] presented a solvability condition for matrix equation (1.2), and derived an expression for the general positive solution to (1.2), which can be generalized into Hilbert C * -module. …”

Section: Elamentioning

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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Section: Elamentioning

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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“…solution to the matrix equation AXA * = B. We remark that Groß (2000) gives another form of a general n.n.d. solution to AXA * = B.…”

Section: Lemmasmentioning

confidence: 98%

Independence distribution-preserving covariance structures for the likelihood ratio test for LXB=0 in the general linear model

Young¹

2015

IJSP

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Research has been ongoing for over fifty years with respect to violating the assumption of i.i.d. observations in the error covariance matrix. However, there exist test statistics and dependency structures for which the sampling distribution of the test statistic is identical to the test statistic's distribution under the assumption of i.i.d. observations. We derive an explicit representation of the general non-i.i.d. error covariance matrix of the general linear model error vector such that the likelihood ratio test statistic for testing certain linear restrictions on the parameter vector is robust against certain forms of dependency and heteroscedasticity. In doing so, we correct two proposed explicit covariance matrix characterizations given in Khatri (1981).

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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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Nonnegative-definite and positive-definite solutions to the matrix equationAXA∗=B – revisited

Cited by 58 publications

References 4 publications

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

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

Independence distribution-preserving covariance structures for the likelihood ratio test for LXB=0 in the general linear model

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

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