The η-anti-Hermitian solution to some classic matrix equations

Liu, Xin

doi:10.1016/j.amc.2017.09.033

Cited by 20 publications

(11 citation statements)

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“…In this case, the split quaternion matrix Equation (13) also has a unique least squares -(anti-)Hermitian solution X = X 1 + X 2 i + X 3 j + X 4 k, which is given as follows:…”

Section: Where Y = ((A ) T a ) † ((A ) T B A )((A ) T A ) † + R A U ± (Rmentioning

confidence: 99%

“…The concept of -(anti-)Hermitian matrix is a more generalized concept comparing with (skew-)Hermitian matrix. This kind of matrices over H have been widely investigated (see, eg, previous studies 1,7,8,13,17,19,21,25 ). For instance, a singular value decomposition for -Hermitian matrices over H, 7 -(anti-)Hermitian solution to quaternion matrix equations such as AX+(AX) * +BYB * +CZC * = D, AX = B, AXB = C, AXA * = B, EXE * +FYF * = H (see, eg, previous studies 8,13,25 ), the least squares -(anti-)Hermitian solution to AXB + CYD = E, AXB + CXD = E, AXB = C (see, eg, previous studies 1,19,21 ), and other kinds of equations and solutions (see, eg, previous studies 2, 3,18,22,23,26 ).…”

Section: Introductionmentioning

confidence: 99%

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Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B

Liu

Zhang

2019

Math Methods in App Sciences

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In the paper, the split quaternion matrix equation AXAη*=B is considered, where the operator Aη* is the η‐conjugate transpose of A, where η∈{i,j,k}. We propose some new real representations, which well exploited the special structures of the original matrices. By using this method, we obtain the necessary and sufficient conditions for AXAη*=B to have X=±Xη* solutions and derive the general expressions of solutions when it is consistent. In addition, we also derive the general expressions of the least squares X=±Xη* solutions to it in case that this matrix equation is not consistent.

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“…In this case, the split quaternion matrix Equation (13) also has a unique least squares -(anti-)Hermitian solution X = X 1 + X 2 i + X 3 j + X 4 k, which is given as follows:…”

Section: Where Y = ((A ) T a ) † ((A ) T B A )((A ) T A ) † + R A U ± (Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B

Liu

Zhang

2019

Math Methods in App Sciences

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“…The singular value decomposition of the -Hermitian matrix was examined in [17]. Very recently, a researcher in [20] determined the skew--Hermitian solution to some significant matrix equations including…”

Section: Introductionmentioning

confidence: 99%

The General Solution of Quaternion Matrix Equation Having η‐Skew‐Hermicity and Its Cramer’s Rule

Rehman

Kyrchei

Ali

et al. 2019

Mathematical Problems in Engineering

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We determine some necessary and sufficient conditions for the existence of the η-skew-Hermitian solution to the following system AX-(AX)η⁎+BYBη⁎+CZCη⁎=D,Y=-Yη⁎,Z=-Zη⁎ over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to establish the main result.

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“…As a special case of ϕ-(skew)-Hermitian matrix, the η-(anti)-Hermitian matrix first arises in widely linear modelling and has important applications over independent component analysis [2] and convergence analysis in statistical signal processing [3]. Recently, it was widely investigated (e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17]). For instance, Horn and Zhang [15] gave a singular value decomposition for η-Hermitian matrix.…”

Section: Introductionmentioning

confidence: 99%

Real Representation Approach to Quaternion Matrix Equation Involving ϕ‐Hermicity

Liu

Huang

2019

Mathematical Problems in Engineering

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For a quaternion matrix A, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.

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The η-anti-Hermitian solution to some classic matrix equations

Cited by 20 publications

References 10 publications

Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B

Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B

The General Solution of Quaternion Matrix Equation Having η‐Skew‐Hermicity and Its Cramer’s Rule

Real Representation Approach to Quaternion Matrix Equation Involving ϕ‐Hermicity

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The η-anti-Hermitian solution to some classic matrix equations

Cited by 20 publications

References 10 publications

Least squares X=±Xη* solutions to split quaternion matrix equation AXAη*=B

Least squares X=±Xη* solutions to split quaternion matrix equation AXAη*=B

The General Solution of Quaternion Matrix Equation Having η‐Skew‐Hermicity and Its Cramer’s Rule

Real Representation Approach to Quaternion Matrix Equation Involving ϕ‐Hermicity

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Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B

Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B