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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 307036, 19 pages
http://dx.doi.org/10.1155/2012/307036
Research Article

The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 9 October 2011; Accepted 17 November 2011

Academic Editor: Qing-WenΒ Wang

Copyright Β© 2012 Shao-Wen Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equations 𝐴1𝑋=𝐢1,𝑋𝐡1=𝐢2,and𝐴3π‘‹π΄βˆ—3=𝐢3. Moreover, formulas of the maximal and minimal ranks of four real matrices 𝑋1,𝑋2,𝑋3, and 𝑋4 in solution 𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜ to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations 𝐴1𝑋=𝐢1,𝑋𝐡1=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3, and𝐴4π‘‹π΄βˆ—4=𝐢4 to have real and complex Hermitian solutions.

1. Introduction

Throughout this paper, we denote the real number field by ℝ; the complex field by β„‚; the set of all π‘šΓ—π‘› matrices over the quaternion algebra ξ€½π‘Žβ„=0+π‘Ž1𝑖+π‘Ž2𝑗+π‘Ž3π‘˜βˆ£π‘–2=𝑗2=π‘˜2=π‘–π‘—π‘˜=βˆ’1,π‘Ž0,π‘Ž1,π‘Ž2,π‘Ž3ξ€Ύβˆˆβ„(1.1) by β„π‘šΓ—π‘›; the identity matrix with the appropriate size by 𝐼; the transpose, the conjugate transpose, the column right space, the row left space of a matrix 𝐴 over ℍ by 𝐴𝑇,π΄βˆ—,β„›(𝐴), 𝒩(𝐴), respectively; the dimension of β„›(𝐴) by dimβ„›(𝐴). By [1], for a quaternion matrix 𝐴,dimβ„›(𝐴)=dim𝒩(𝐴). dimβ„›(𝐴) is called the rank of a quaternion matrix 𝐴 and denoted by π‘Ÿ(𝐴). The Moore-Penrose inverse of matrix 𝐴 over ℍ by 𝐴† which satisfies four Penrose equations 𝐴𝐴†𝐴=𝐴,𝐴†𝐴𝐴†=𝐴†,(𝐴𝐴†)βˆ—=𝐴𝐴†, and (𝐴†𝐴)βˆ—=𝐴†𝐴. In this case, 𝐴† is unique and (𝐴†)βˆ—=(π΄βˆ—)†. Moreover, 𝑅𝐴 and 𝐿𝐴 stand for the two projectors 𝐿𝐴=πΌβˆ’π΄β€ π΄, and 𝑅𝐴=πΌβˆ’π΄π΄β€  induced by 𝐴. Clearly, 𝑅𝐴 and 𝐿𝐴 are idempotent and satisfies (𝑅𝐴)βˆ—=𝑅𝐴,(𝐿𝐴)βˆ—=𝐿𝐴,𝑅𝐴=πΏπ΄βˆ—, and π‘…π΄βˆ—=𝐿𝐴.

Hermitian solutions to some matrix equations were investigated by many authors. In 1976, Khatri and Mitra [2] gave necessary and sufficient conditions for the existence of the Hermitian solutions to the matrix equations 𝐴𝑋=𝐡,𝐴𝑋𝐡=𝐢 and 𝐴1𝑋=𝐢1,𝑋𝐡2=𝐢2,(1.2) over the complex field β„‚, and presented explicit expressions for the general Hermitian solutions to them by generalized inverses when the solvability conditions were satisfied. Matrix equation that has symmetric patterns with Hermitian solutions appears in some application areas, such as vibration theory, statistics, and optimal control theory ([3–7]). Groß in [8], and Liu et al. in [9] gave the solvability conditions for Hermitian solution and its expressions of π΄π‘‹π΄βˆ—=𝐡(1.3) over β„‚ in terms of generalized inverses, respectively. In [10], Tian and Liu established the solvability conditions for 𝐴3π‘‹π΄βˆ—3=𝐢3,𝐴4π‘‹π΄βˆ—4=𝐢4(1.4) to have a common Hermitian solution over β„‚ by the ranks of coefficient matrices. In [11], Tian derived the general common Hermitian solution of (1.4). Wang and Wu in [12] gave some necessary and sufficient conditions for the existence of the common Hermitian solution to equations 𝐴1𝑋=𝐢1,𝑋𝐡2=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3,𝐴(1.5)1𝑋=𝐢1,𝑋𝐡2=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3,𝐴4π‘‹π΄βˆ—4=𝐢4,(1.6) for operators between Hilbert πΆβˆ—-modules by generalized inverses and range inclusion of matrices.

As is known to us, extremal ranks of some matrix expressions can be used to characterize nonsingularity, rank invariance, range inclusion of the corresponding matrix expressions, as well as solvability conditions of matrix equations ([4, 7, 9–24]). Real matrices and its extremal ranks in solutions to some complex matrix equation have been investigated by Tian and Liu ([9, 13–15]). Tian [13] gave the maximal and minimal ranks of two real matrices 𝑋0 and 𝑋1 in solution 𝑋=𝑋0+𝑖𝑋1 to 𝐴𝑋𝐡=𝐢 over β„‚ with its applications. Liu et al. [9] derived the maximal and minimal ranks of the two real matrices 𝑋0 and 𝑋1 in a Hermitian solution 𝑋=𝑋0+𝑖𝑋1 of (1.3), where π΅βˆ—=𝐡. In order to investigate the real and complex solutions to quaternion matrix equations, Wang and his partners have been studying the real matrices in solutions to some quaternion matrix equations such as 𝐴𝑋𝐡=𝐢, 𝐴1𝑋𝐡1=𝐢1,𝐴2𝑋𝐡2=𝐢2,π΄π‘‹π΄βˆ—+π΅π‘‹π΅βˆ—=𝐢,(1.7) recently ([24–27]). To our knowledge, the necessary and sufficient conditions for (1.5) over ℍ to have the real and complex Hermitian solutions have not been given so far. Motivated by the work mentioned above, we in this paper investigate the real and complex Hermitian solutions to system (1.5) over ℍ and its applications.

This paper is organized as follows. In Section 2, we first derive formulas of extremal ranks of four real matrices 𝑋1,𝑋2,𝑋3, and 𝑋4 in quaternion solution 𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜ to (1.5) over ℍ, then give necessary and sufficient conditions for (1.5) over ℍ to have real and complex solutions as well as the expressions of the real and complex solutions. As applications, we in Section 3 establish necessary and sufficient conditions for (1.6) over ℍ to have real and complex solutions.

2. The Real and Complex Hermitian Solutions to System (1.5) Over ℍ

In this section, we first give a solvability condition and an expression of the general Hermitian solution to (1.5) over ℍ, then consider the maximal and minimal ranks of four real matrices 𝑋1,𝑋2,𝑋3, and 𝑋4 in solution 𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜ to (1.5) over ℍ, last, investigate the real and complex Hermitian solutions to (1.5) over ℍ.

For an arbitrary matrix 𝑀𝑑=𝑀𝑑1+𝑀𝑑2𝑖+𝑀𝑑3𝑗+𝑀𝑑4π‘˜βˆˆβ„π‘šΓ—π‘› where 𝑀𝑑1,𝑀𝑑2,𝑀𝑑3, and 𝑀𝑑4 are real matrices, we define a map πœ™(β‹…) from β„π‘šΓ—π‘› to ℝ4π‘šΓ—4𝑛 by πœ™ξ€·π‘€π‘‘ξ€Έ=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘€π‘‘1𝑀𝑑2𝑀𝑑3𝑀𝑑4βˆ’π‘€π‘‘2𝑀𝑑1𝑀𝑑4βˆ’π‘€π‘‘3βˆ’π‘€π‘‘3βˆ’π‘€π‘‘4𝑀𝑑1𝑀𝑑2βˆ’π‘€π‘‘4𝑀𝑑3βˆ’π‘€π‘‘2𝑀𝑑1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(2.1) By (2.1), it is easy to verify that πœ™(β‹…) satisfies the following properties.(a)𝑀=π‘β‡”πœ™(𝑀)=πœ™(𝑁). (b)πœ™(π‘˜π‘€+𝑙𝑁)=π‘˜πœ™(𝑀)+π‘™πœ™(𝑁),πœ™(𝑀𝑁)=πœ™(𝑀)πœ™(𝑁),π‘˜,π‘™βˆˆβ„.(c)πœ™(π‘€βˆ—)=πœ™π‘‡(𝑀), πœ™(𝑀†)=πœ™β€ (𝑀). (d)πœ™(𝑀)=π‘‡π‘šβˆ’1πœ™(𝑀)𝑇𝑛=π‘…π‘šβˆ’1πœ™(𝑀)𝑅𝑛=π‘†π‘šβˆ’1πœ™(𝑀)𝑆𝑛, where 𝑑=π‘š,𝑛, 𝑇𝑑=⎑⎒⎒⎒⎒⎒⎒⎣0βˆ’πΌπ‘‘πΌ00𝑑000000𝐼𝑑00βˆ’πΌπ‘‘0⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑅𝑑=⎑⎒⎒⎒⎒⎒⎒⎣00βˆ’πΌπ‘‘0000βˆ’πΌπ‘‘πΌπ‘‘0000πΌπ‘‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦00,𝑆𝑑=⎑⎒⎒⎒⎒⎒⎒⎣000βˆ’πΌπ‘‘00𝐼𝑑00βˆ’πΌπ‘‘πΌ00π‘‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦000.(2.2)(e)π‘Ÿ[πœ™(𝑀)]=4π‘Ÿ(𝑀). (f)π‘€βˆ—=π‘€β‡”πœ™π‘‡(𝑀)=πœ™(𝑀), π‘€βˆ—=βˆ’π‘€β‡”πœ™π‘‡(𝑀)=βˆ’πœ™(𝑀).

The following lemmas provide us with some useful results over β„‚, which can be generalized to ℍ.

Lemma 2.1 (see [2, Lemma  2.1]). Let π΄βˆˆβ„π‘šΓ—π‘›,𝐡=π΅βˆ—βˆˆβ„π‘šΓ—π‘š be known, π‘‹βˆˆβ„π‘›Γ—π‘› unknown; then the system (1.3) has a Hermitian solution if and only if 𝐴𝐴†𝐡=𝐡.(2.3) In that case, the general Hermitian solution of (1.3) can be expressed as 𝑋=π΄β€ π΅ξ€·π΄β€ ξ€Έβˆ—+𝐿𝐴𝑉+π‘‰βˆ—πΏπ΄,(2.4) where 𝑉 is arbitrary matrix over ℍ with compatible size.

Lemma 2.2 (see [12, Corollary  3.4]). Let 𝐴1,𝐢1βˆˆβ„π‘šΓ—π‘›;𝐡1,𝐢2βˆˆβ„π‘›Γ—π‘ ;𝐴3βˆˆβ„π‘ŸΓ—π‘›;𝐢3βˆˆβ„π‘ŸΓ—π‘Ÿ be known, π‘‹βˆˆβ„π‘›Γ—π‘› unknown, and 𝐹=π΅βˆ—1𝐿𝐴1,𝑀=𝑆𝐿𝐹,𝑆=𝐴3𝐿𝐴1,𝐷=πΆβˆ—2βˆ’π΅βˆ—1𝐴†1𝐢1,𝐽=𝐴†1𝐢1+𝐹†𝐷,𝐺=𝐢3βˆ’π΄3(𝐽+𝐿𝐴1πΏβˆ—πΉπ½βˆ—)π΄βˆ—3,𝐢3=πΆβˆ—3; then the system (1.5) have a Hermitian solution if and only if 𝐴1𝐢2=𝐢1𝐡1,𝐴1πΆβˆ—1=𝐢1π΄βˆ—1,π΅βˆ—1𝐢2=πΆβˆ—2𝐡1,𝑅𝐴1𝐢1=0,𝑅𝐹𝐷=0,𝑅𝑀𝐺=0.(2.5) In that case, the general Hermitian solution of (1.5) can be expressed as 𝑋=𝐽+𝐿𝐴1πΏπΉπ½βˆ—+𝐿𝐴1πΏπΉπ‘€β€ πΊξ€·π‘€β€ ξ€Έβˆ—πΏπΉπΏπ΄1+𝐿𝐴1𝐿𝐹𝐿𝑀𝑉𝐿𝐹𝐿𝐴1+𝐿𝐴1πΏπΉπ‘‰βˆ—πΏπ‘€πΏπΉπΏπ΄1,(2.6) where 𝑉is arbitrary matrix over ℍ with compatible size.

Lemma 2.3 (see [21, Lemma  2.4]). Let π΄βˆˆβ„π‘šΓ—π‘›,π΅βˆˆβ„π‘šΓ—π‘˜,πΆβˆˆβ„π‘™Γ—π‘›,π·βˆˆβ„π‘—Γ—π‘˜, and πΈβˆˆβ„π‘™Γ—π‘–. Then they satisfy the following rank equalities.(a)π‘Ÿ(𝐢𝐿𝐴)=π‘Ÿπ΄πΆξ€»βˆ’π‘Ÿ(𝐴). (b)π‘Ÿ[𝐡𝐴𝐿𝐢]ξ€Ί=π‘Ÿπ΅π΄0πΆξ€»βˆ’π‘Ÿ(𝐢). (c)π‘Ÿξ€ΊπΆπ‘…π΅π΄ξ€»ξ€Ί=π‘ŸπΆ0π΄π΅ξ€»βˆ’π‘Ÿ(𝐡). (d)π‘Ÿξ‚ƒπ΄π΅πΏπ·π‘…πΈπΆ0=π‘Ÿπ΄π΅0𝐢0𝐸0𝐷0ξ‚„βˆ’π‘Ÿ(𝐷)βˆ’π‘Ÿ(𝐸).

Lemma 2.3 plays an important role in simplifying ranks of various block matrices.

Lemma 2.4 (see [11, Theorem  4.1, Corollary  4.2]). Let 𝐴=Β±π΄βˆ—βˆˆβ„π‘šΓ—π‘š,π΅βˆˆβ„π‘šΓ—π‘›, andπΆβˆˆβ„π‘Γ—π‘š be given; then maxπ‘‹βˆˆβ„π‘›Γ—π‘π‘Ÿξ€Ίπ΄βˆ’π΅π‘‹πΆβˆ“(𝐡𝑋𝐢)βˆ—ξ€»βŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ‚ƒ=minπ΄π΅πΆβˆ—ξ‚„βŽ‘βŽ’βŽ’βŽ£π΅,π‘Ÿπ΄π΅βˆ—0⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣,π‘Ÿπ΄πΆβˆ—βŽ€βŽ₯βŽ₯⎦⎫βŽͺ⎬βŽͺ⎭,𝐢0minπ‘‹βˆˆβ„π‘›Γ—π‘π‘Ÿξ€Ίπ΄βˆ’π΅π‘‹πΆβˆ“(𝐡𝑋𝐢)βˆ—ξ€»ξ‚ƒ=2π‘Ÿπ΄π΅πΆβˆ—ξ‚„ξ€½π‘ +max1,𝑠2ξ€Ύ,(2.7) where 𝑠1⎑⎒⎒⎣𝐡=π‘Ÿπ΄π΅βˆ—0⎀βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£βˆ’2π‘Ÿπ΄π΅πΆβˆ—π΅βˆ—βŽ€βŽ₯βŽ₯⎦,𝑠002⎑⎒⎒⎣=π‘Ÿπ΄πΆβˆ—βŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎣𝐢0βˆ’2π‘Ÿπ΄π΅πΆβˆ—βŽ€βŽ₯βŽ₯⎦.𝐢00(2.8) If β„›(𝐡)βŠ†β„›(πΆβˆ—), maxπ‘‹βˆˆβ„π‘›Γ—π‘π‘Ÿξ€Ίπ΄βˆ’π΅π‘‹πΆβˆ’(𝐡𝑋𝐢)βˆ—ξ€»βŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ‚ƒ=minπ΄πΆβˆ—ξ‚„βŽ‘βŽ’βŽ’βŽ£π΅,π‘Ÿπ΄π΅βˆ—0⎀βŽ₯βŽ₯⎦⎫βŽͺ⎬βŽͺ⎭,(2.9)minπ‘‹βˆˆβ„π‘›Γ—π‘π‘Ÿξ€Ίπ΄βˆ’π΅π‘‹πΆβˆ’(𝐡𝑋𝐢)βˆ—ξ€»ξ‚ƒ=2π‘Ÿπ΄πΆβˆ—ξ‚„βŽ‘βŽ’βŽ’βŽ£π΅+π‘Ÿπ΄π΅βˆ—0⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’2π‘Ÿπ΄π΅πΆ0.(2.10)

Lemma 2.5 (see [28, Theorem  3.1]). Let π΄βˆˆβ„π‘šΓ—π‘›,𝐡1βˆˆβ„π‘šΓ—π‘1,𝐡3βˆˆβ„π‘šΓ—π‘3,𝐡4βˆˆβ„π‘šΓ—π‘4,𝐢2βˆˆβ„π‘ž2×𝑛,𝐢3βˆˆβ„π‘ž3×𝑛, and 𝐢4βˆˆβ„π‘ž4×𝑛 be given. Then the matrix equation 𝐡1𝑋1+𝑋2𝐢2+𝐡3𝑋3𝐢3+𝐡4𝑋4𝐢4=𝐴(2.11) is consistent if and only if π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π΄π΅1𝐢20𝐢30𝐢40⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎒⎒⎣=π‘Ÿ0𝐡1𝐢20𝐢30𝐢40⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎣,π‘Ÿπ΄π΅1𝐡3𝐡4𝐢2⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣000=π‘Ÿ0𝐡1𝐡3𝐡4𝐢2⎀βŽ₯βŽ₯⎦,π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£000𝐴𝐡1𝐡3𝐢2𝐢004⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣00=π‘Ÿ0𝐡1𝐡3𝐢2𝐢004⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣00,π‘Ÿπ΄π΅1𝐡4𝐢2𝐢003⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣00=π‘Ÿ0𝐡1𝐡4𝐢2𝐢003⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.00(2.12)

Theorem 2.6. System (1.5) has a Hermitian solution over ℍ if and only if the system of matrix equations πœ™ξ€·π΄1π‘Œξ€Έξ€·π‘–π‘—ξ€Έ4Γ—4𝐢=πœ™1ξ€Έ,ξ€·π‘Œπ‘–π‘—ξ€Έ4Γ—4πœ™ξ€·π΅1𝐢=πœ™2𝐴,πœ™3π‘Œξ€Έξ€·π‘–π‘—ξ€Έ4Γ—4πœ™π‘‡ξ€·π΄3𝐢=πœ™3ξ€Έ,𝑖,𝑗=1,2,3,4,(2.13) has a symmetric solution over ℝ. In that case, the general Hermitian solution of (1.5) over ℍ can be written as 𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜=14ξ€·π‘Œ11+π‘Œ22+π‘Œ33+π‘Œ44ξ€Έ+14ξ€·π‘Œ12βˆ’π‘Œπ‘‡12+π‘Œ34βˆ’π‘Œπ‘‡34𝑖+14ξ€·π‘Œ13βˆ’π‘Œπ‘‡13+π‘Œπ‘‡24βˆ’π‘Œ24ξ€Έ1𝑗+4ξ€·π‘Œ14βˆ’π‘Œπ‘‡14+π‘Œ23βˆ’π‘Œπ‘‡23ξ€Έπ‘˜,(2.14) where π‘Œπ‘‘π‘‘=π‘Œπ‘‡π‘‘π‘‘;𝑑=1,2,3,4;π‘Œπ‘‡1𝑗=π‘Œπ‘—1;𝑗=2,3,4;π‘Œπ‘‡2𝑗=π‘Œπ‘—2;𝑗=3,4;π‘Œπ‘‡34=π‘Œ43 are the general solutions of (2.13) over ℝ. Written in an explicit form, 𝑋1,𝑋2,𝑋3, and 𝑋4 in (2.14) are 𝑋1=14𝑃1πœ™ξ€·π‘‹0𝑃𝑇1+14𝑃2πœ™ξ€·π‘‹0𝑃𝑇2+14𝑃3πœ™ξ€·π‘‹0𝑃𝑇3+14𝑃4πœ™ξ€·π‘‹0𝑃𝑇4+𝑃1,𝑃2,𝑃3,𝑃4ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+𝑃1,𝑃2,𝑃3,𝑃4ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)π‘‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑋(2.15)2=14𝑃1πœ™ξ€·π‘‹0𝑃𝑇2βˆ’14𝑃2πœ™ξ€·π‘‹0𝑃𝑇1+14𝑃3πœ™ξ€·π‘‹0𝑃𝑇4βˆ’14𝑃4πœ™ξ€·π‘‹0𝑃𝑇3+𝑃1,βˆ’π‘ƒ2,𝑃3,βˆ’π‘ƒ4ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+𝑃2,𝑃1,𝑃4,𝑃3ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)π‘‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1βˆ’πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3βˆ’πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑋(2.16)3=14𝑃1πœ™ξ€·π‘‹0𝑃𝑇3βˆ’14𝑃3πœ™ξ€·π‘‹0𝑃𝑇1+14𝑃4πœ™ξ€·π‘‹0𝑃𝑇2βˆ’14𝑃2πœ™ξ€·π‘‹0𝑃𝑇4+𝑃1,βˆ’π‘ƒ3,𝑃4,βˆ’π‘ƒ2ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+𝑃3,𝑃1,𝑃2,𝑃4ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)π‘‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1βˆ’πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4βˆ’πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑋(2.17)4=14𝑃1πœ™ξ€·π‘‹0𝑃𝑇4βˆ’14𝑃4πœ™ξ€·π‘‹0𝑃𝑇1+14𝑃2πœ™ξ€·π‘‹0𝑃𝑇3βˆ’14𝑃3πœ™ξ€·π‘‹0𝑃𝑇2+𝑃1,βˆ’π‘ƒ4,𝑃2,βˆ’π‘ƒ3ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+𝑃4,𝑃1,𝑃3,𝑃2ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)π‘‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1βˆ’πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2βˆ’πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(2.18) where 𝑃1=𝐼𝑛,0,0,0,𝑃2=ξ€Ί0,𝐼𝑛,0,0,𝑃3=ξ€Ί0,0,𝐼𝑛,0,𝑃4=ξ€Ί0,0,0,𝐼𝑛,(2.19)πœ™(𝑋0) is a particular symmetric solution to (2.13), and 𝑉 is arbitrary real matrices with compatible sizes.

Proof. Suppose that (1.5) has a Hermitian solution 𝑋 over ℍ. Applying properties (a) and (b) of πœ™(β‹…) to (1.5) yields πœ™ξ€·π΄1ξ€Έπœ™ξ€·πΆ(𝑋)=πœ™1𝐡,πœ™(𝑋)πœ™2𝐢=πœ™2𝐴,πœ™3ξ€Έπœ™(𝑋)πœ™π‘‡ξ€·π΄3𝐢=πœ™3ξ€Έ,(2.20) implying that πœ™(𝑋) is a real symmetric solution to (2.13).
Conversely, suppose that (2.13) has a real symmetric solution ξξπ‘Œπ‘Œ=𝑇=ξ€·π‘Œπ‘–π‘—ξ€Έ4Γ—4,𝑖,𝑗=1,2,3,4.(2.21) That is, πœ™ξ€·π΄1ξ€Έξξ€·πΆπ‘Œ=πœ™1ξ€Έ,ξξ€·π΅π‘Œπœ™2𝐢=πœ™2𝐴,πœ™3ξ€Έξπ‘Œπœ™π‘‡ξ€·π΄3𝐢=πœ™3ξ€Έ,(2.22) then by property (𝑑) of πœ™(β‹…), π‘‡π‘šβˆ’1πœ™ξ€·π΄1ξ€Έπ‘‡π‘›ξπ‘Œ=π‘‡π‘šβˆ’1πœ™ξ€·πΆ1𝑇𝑛,ξπ‘Œπ‘‡π‘›βˆ’1πœ™ξ€·π΅2𝑇𝑠=π‘‡π‘›βˆ’1πœ™ξ€·πΆ2𝑇𝑠,π‘‡π‘Ÿβˆ’1πœ™ξ€·π΄3ξ€Έπ‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1πœ™π‘‡ξ€·π΄3ξ€Έπ‘‡π‘Ÿ=π‘‡π‘Ÿβˆ’1πœ™ξ€·πΆ3ξ€Έπ‘‡π‘Ÿ,π‘…π‘šβˆ’1πœ™ξ€·π΄1ξ€Έπ‘…π‘›ξπ‘Œ=π‘…π‘šβˆ’1πœ™ξ€·πΆ1𝑅𝑛,ξπ‘Œπ‘…π‘›βˆ’1πœ™ξ€·π΅2𝑅𝑠=π‘…π‘›βˆ’1πœ™ξ€·πΆ2𝑅𝑠,π‘…π‘Ÿβˆ’1πœ™ξ€·π΄3ξ€Έπ‘…π‘›ξπ‘Œπ‘…π‘›πœ™π‘‡ξ€·π΄3ξ€Έπ‘…π‘Ÿ=π‘…π‘Ÿβˆ’1πœ™ξ€·πΆ3ξ€Έπ‘…π‘Ÿ,π‘†π‘šβˆ’1πœ™ξ€·π΄1ξ€Έπ‘†π‘›ξπ‘Œ=π‘†π‘šβˆ’1πœ™ξ€·πΆ1𝑆𝑛,ξπ‘Œπ‘†π‘›βˆ’1πœ™ξ€·π΅2𝑆𝑠=π‘†π‘›βˆ’1πœ™ξ€·πΆ2𝑆𝑠,π‘†π‘Ÿβˆ’1πœ™ξ€·π΄3ξ€Έπ‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1πœ™π‘‡ξ€·π΄3ξ€Έπ‘†π‘Ÿ=π‘†π‘Ÿβˆ’1πœ™ξ€·πΆ3ξ€Έπ‘†π‘Ÿ.(2.23) Hence, πœ™ξ€·π΄1ξ€Έπ‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1𝐢=πœ™1ξ€Έ,π‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1πœ™ξ€·π΅2𝐢=πœ™2𝐴,πœ™3ξ€Έπ‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1πœ™π‘‡ξ€·π΄3𝐢=πœ™3ξ€Έ,πœ™ξ€·π΄1ξ€Έπ‘…π‘›ξπ‘Œπ‘…π‘›βˆ’1𝐢=πœ™1ξ€Έ,π‘…π‘›ξπ‘Œπ‘…π‘›βˆ’1πœ™ξ€·π΅2𝐢=πœ™2𝐴,πœ™3ξ€Έπ‘…π‘›ξπ‘Œπ‘…π‘›βˆ’1πœ™π‘‡ξ€·π΄3𝐢=πœ™3ξ€Έ,πœ™ξ€·π΄1ξ€Έπ‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1𝐢=πœ™1ξ€Έ,π‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1πœ™ξ€·π΅2𝐢=πœ™2𝐴,πœ™3ξ€Έπ‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1πœ™π‘‡ξ€·π΄3𝐢=πœ™3ξ€Έ,(2.24) implying that π‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1,π‘…π‘›ξπ‘Œπ‘…π‘›βˆ’1, and π‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1 are also symmetric solutions of (2.13). Thus, 14ξ‚€ξπ‘Œ+π‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1+π‘…π‘›ξπ‘Œπ‘…π‘›βˆ’1+π‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1(2.25) is a symmetric solution of (2.13), where ξπ‘Œ+π‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1+π‘…π‘›ξπ‘Œπ‘…π‘›βˆ’1+π‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1=ξ‚€ξ‚‹π‘Œπ‘–π‘—ξ‚4Γ—4ξ‚‹π‘Œ,𝑖=1,2,3,4,11=π‘Œ11+π‘Œ22+π‘Œ33+π‘Œ44,ξ‚‹π‘Œ12=π‘Œ12βˆ’π‘Œπ‘‡12+π‘Œ34βˆ’π‘Œπ‘‡34,ξ‚‹π‘Œ13=π‘Œ13βˆ’π‘Œπ‘‡13+π‘Œπ‘‡24βˆ’π‘Œ24,ξ‚‹π‘Œ14=π‘Œ14βˆ’π‘Œπ‘‡14+π‘Œ23βˆ’π‘Œπ‘‡23,ξ‚‹π‘Œ21=π‘Œπ‘‡12βˆ’π‘Œ12+π‘Œπ‘‡34βˆ’π‘Œ34,ξ‚‹π‘Œ22=π‘Œ11+π‘Œ22+π‘Œ33+π‘Œ44,ξ‚‹π‘Œ23=π‘Œ14βˆ’π‘Œπ‘‡14+π‘Œ23βˆ’π‘Œπ‘‡23,ξ‚‹π‘Œ24=π‘Œ13βˆ’π‘Œπ‘‡13+π‘Œπ‘‡24βˆ’π‘Œ24,ξ‚‹π‘Œ31=π‘Œπ‘‡13βˆ’π‘Œ13+π‘Œ24βˆ’π‘Œπ‘‡24,ξ‚‹π‘Œ32=π‘Œ14βˆ’π‘Œπ‘‡14+π‘Œ23βˆ’π‘Œπ‘‡23,ξ‚‹π‘Œ33=π‘Œ11+π‘Œ22+π‘Œ33+π‘Œ44,ξ‚‹π‘Œ34=π‘Œ12βˆ’π‘Œπ‘‡12+π‘Œ34βˆ’π‘Œπ‘‡34,ξ‚‹π‘Œ41=π‘Œπ‘‡14βˆ’π‘Œ14+π‘Œπ‘‡23βˆ’π‘Œ23,ξ‚‹π‘Œ42=π‘Œ13βˆ’π‘Œπ‘‡13+π‘Œπ‘‡24βˆ’π‘Œ24,ξ‚‹π‘Œ43=π‘Œ12βˆ’π‘Œπ‘‡12+π‘Œ34βˆ’π‘Œπ‘‡34,ξ‚‹π‘Œ44=π‘Œ11+π‘Œ22+π‘Œ33+π‘Œ44.(2.26) Let 1𝑋=4ξ€·π‘Œ11+π‘Œ22+π‘Œ33+π‘Œ44ξ€Έ+14ξ€·π‘Œ12βˆ’π‘Œπ‘‡12+π‘Œ34βˆ’π‘Œπ‘‡34𝑖+14ξ€·π‘Œ13βˆ’π‘Œπ‘‡13+π‘Œπ‘‡24βˆ’π‘Œ24ξ€Έ1𝑗+4ξ€·π‘Œ14βˆ’π‘Œπ‘‡14+π‘Œ23βˆ’π‘Œπ‘‡23ξ€Έπ‘˜.(2.27) Then by (2.1), πœ™ξ‚€ξπ‘‹ξ‚=14ξ‚€ξπ‘Œ+π‘‡π‘›ξπ‘Œπ‘‡π‘›βˆ’1+π‘…π‘›ξπ‘Œπ‘…π‘›βˆ’1+π‘†π‘›ξπ‘Œπ‘†π‘›βˆ’1.(2.28) Hence, by the property (a), we know that 𝑋 is a Hermitian solution of (1.5). Observe that π‘Œπ‘–π‘—,𝑖,𝑗=1,2,3,4, in (2.13) can be written as π‘Œπ‘–π‘—=π‘ƒπ‘–ξπ‘Œπ‘ƒπ‘‡π‘—.(2.29) From Lemma 2.2, the general Hermitian solution to (2.13) can be written as ξξ€·π‘‹π‘Œ=πœ™0ξ€Έ+4πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰πΏπœ™(𝐴1)πΏπœ™(𝐹)+4πΏπœ™(𝐹)πΏπœ™(𝐴1)π‘‰π‘‡πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1),(2.30) where π‘‰βˆˆβ„ is arbitrary. Hence, π‘Œπ‘–π‘—=π‘ƒπ‘–πœ™ξ€·π‘‹0𝑃𝑇𝑗+4π‘ƒπ‘–πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰πΏπœ™(𝐴1)πΏπœ™(𝐹)𝑃𝑇𝑗+4π‘ƒπ‘–πΏπœ™(𝐹)πΏπœ™(𝐴1)π‘‰π‘‡πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇𝑗,(2.31) where 𝑖,𝑗=1,2,3,4, substituting them into (2.14), yields the four real matrices 𝑋1,𝑋2,𝑋3, and 𝑋4 in (2.15)–(2.18).

Now we consider the maximal and minimal ranks of four real matrices 𝑋1,𝑋2,𝑋3, and 𝑋4 in solution 𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜ to (1.5) over ℍ.

Theorem 2.7. Suppose that system (1.5) over ℍ has a Hermitian solution, and 𝐴1=𝐴11+𝐴12𝑖+𝐴13𝑗+𝐴14π‘˜,𝐢1=𝐢11+𝐢12𝑖+𝐢13𝑗+𝐢14π‘˜βˆˆβ„π‘šΓ—π‘›π΅1=𝐡11+𝐡12𝑖+𝐡13𝑗+𝐡14π‘˜,𝐢2=𝐢21+𝐢22𝑖+𝐢23𝑗+𝐢24π‘˜βˆˆβ„π‘›Γ—π‘ ,𝐴3=𝐴31+𝐴32𝑖+𝐴33𝑗+𝐴34β„π‘˜βˆˆπ‘ŸΓ—π‘›,𝐢3=𝐢31+𝐢32𝑖+𝐢33𝑗+𝐢34π‘˜βˆˆβ„π‘ŸΓ—π‘Ÿπ‘†1=⎧βŽͺ⎨βŽͺβŽ©π‘‹1βˆˆβ„π‘›Γ—π‘›βˆ£π΄1𝑋=𝐢1,𝑋𝐡1=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜βŽ«βŽͺ⎬βŽͺ⎭,𝑆2=⎧βŽͺ⎨βŽͺβŽ©π‘‹2βˆˆβ„π‘Γ—π‘žβˆ£π΄1𝑋=𝐢1,𝑋𝐡1=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜βŽ«βŽͺ⎬βŽͺ⎭,𝑆3=⎧βŽͺ⎨βŽͺβŽ©π‘‹3βˆˆβ„π‘Γ—π‘žβˆ£π΄1𝑋=𝐢1,𝑋𝐡1=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜βŽ«βŽͺ⎬βŽͺ⎭,𝑆4=⎧βŽͺ⎨βŽͺβŽ©π‘‹4βˆˆβ„π‘Γ—π‘žβˆ£π΄1𝑋=𝐢1,𝑋𝐡1=𝐢2,𝐴3π‘‹π΄βˆ—3=𝐢3𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜βŽ«βŽͺ⎬βŽͺ⎭,𝐿21=⎑⎒⎒⎒⎒⎒⎒⎣𝐢21𝐢22𝐢23𝐢24⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐿11=⎑⎒⎒⎒⎒⎒⎒⎣𝐢11βˆ’πΆ12βˆ’πΆ13βˆ’πΆ14⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑀31=⎑⎒⎒⎒⎒⎒⎒⎣𝐴32𝐴33𝐴34𝐴31𝐴34βˆ’π΄33βˆ’π΄34𝐴31𝐴32𝐴33βˆ’π΄32𝐴31⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑀11=⎑⎒⎒⎒⎒⎒⎒⎣𝐴12𝐴13𝐴14𝐴11𝐴14βˆ’π΄13βˆ’π΄14𝐴11𝐴12𝐴13βˆ’π΄12𝐴11⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑀12=⎑⎒⎒⎒⎒⎒⎒⎣𝐴11𝐴13𝐴14βˆ’π΄12𝐴14βˆ’π΄13βˆ’π΄13𝐴11𝐴12βˆ’π΄14βˆ’π΄12𝐴11⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑀13=⎑⎒⎒⎒⎒⎒⎒⎣𝐴11𝐴12𝐴14βˆ’π΄12𝐴11βˆ’π΄13βˆ’π΄13βˆ’π΄14𝐴12βˆ’π΄14𝐴13𝐴11⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑀14=⎑⎒⎒⎒⎒⎒⎒⎣𝐴11𝐴12𝐴13βˆ’π΄12𝐴11𝐴14βˆ’π΄13βˆ’π΄14𝐴11βˆ’π΄14𝐴13βˆ’π΄12⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑁11=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’π΅12𝐡11𝐡14βˆ’π΅13βˆ’π΅13βˆ’π΅14𝐡11𝐡12βˆ’π΅14𝐡13βˆ’π΅12𝐡11⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑁12=⎑⎒⎒⎒⎒⎣𝐡11𝐡12𝐡13𝐡14βˆ’π΅13βˆ’π΅14𝐡11𝐡12βˆ’π΅14𝐡13βˆ’π΅12𝐡11⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑁13=⎑⎒⎒⎒⎒⎣𝐡11𝐡12𝐡13𝐡14βˆ’π΅12𝐡11𝐡14βˆ’π΅13βˆ’π΅14𝐡13βˆ’π΅12𝐡11⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑁14=⎑⎒⎒⎒⎒⎣𝐡11𝐡12𝐡13𝐡14βˆ’π΅12𝐡11𝐡14βˆ’π΅13βˆ’π΅13βˆ’π΅14𝐡11𝐡12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(2.32) Then the maximal and minimal ranks of 𝑋𝑖,𝑖=1,2,3,4, in Hermitian solution 𝑋=𝑋1+𝑋2𝑖+𝑋3𝑗+𝑋4π‘˜ to (1.5) are given bymaxπ‘‹π‘–βˆˆπ‘†π‘–π‘Ÿξ€·π‘‹π‘–ξ€Έξ€½π‘‘=min1𝑖,,𝑑(2.33)minπ‘‹π‘–βˆˆπ‘†π‘–π‘Ÿξ€·π‘‹π‘–ξ€Έ=2𝑑1𝑖+π‘‘βˆ’2𝑑2𝑖,(2.34) where 𝑑1π‘–βŽ‘βŽ’βŽ’βŽ£πΏ=π‘Ÿ21𝑁𝑇1𝑖𝐿11𝑀1π‘–βŽ€βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£π΅βˆ’4π‘Ÿβˆ—1𝐴1⎀βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎒⎒⎣+𝑛,𝑑=π‘Ÿ0𝑀𝑇31𝑁11𝑀𝑇11𝑀31πœ™ξ€·πΆ3ξ€Έπœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1𝑁𝑇11πœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝑀11πœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄βˆ’8π‘Ÿ3π΅βˆ—1𝐴1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‘+2𝑛,2π‘–βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£=π‘Ÿ0𝑁1𝑖𝑀𝑇1𝑖𝑀31πœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1𝑁𝑇11πœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝑀11πœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄βˆ’4π‘Ÿ3π΅βˆ—1𝐴1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£π΅βˆ’4π‘Ÿβˆ—1𝐴1⎀βŽ₯βŽ₯⎦+2𝑛.(2.35)

Proof. We only prove the case that 𝑖=1. Similarly, we can get the results that 𝑖=2,3,4. Let 14𝑃1πœ™ξ€·π‘‹0𝑃𝑇1+14𝑃2πœ™ξ€·π‘‹0𝑃𝑇2+14𝑃3πœ™ξ€·π‘‹0𝑃𝑇3+14𝑃4πœ™ξ€·π‘‹0𝑃𝑇4𝑃=𝐴,1,𝑃2,𝑃3,𝑃4ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)⎑⎒⎒⎒⎒⎒⎒⎣𝐿=𝐡,πœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=𝐢;(2.36) note that 𝐿𝑀 is Hermtian; then πΏπœ™(𝑀) is symmetric; hence (2.15) can be written as 𝑋1=𝐴+𝐡𝑉𝐢+(𝐡𝑉𝐢)βˆ—.(2.37) Note that 𝐴=π΄βˆ— and β„›(𝐡)βŠ†β„›(πΆβˆ—); applying (2.9) and (2.10) to (2.37) yields max𝑋1βˆˆπ‘†1π‘Ÿξ€·π‘‹1ξ€ΈβŽ§βŽͺ⎨βŽͺβŽ©π‘Ÿξ€Ί=min𝐴,πΆβˆ—ξ€»βŽ‘βŽ’βŽ’βŽ£π΅,π‘Ÿπ΄π΅βˆ—0⎀βŽ₯βŽ₯⎦⎫βŽͺ⎬βŽͺ⎭,(2.38)min𝑋1βˆˆπ‘†1π‘Ÿξ€·π‘‹1ξ€Έξ€Ί=2π‘Ÿπ΄,πΆβˆ—ξ€»βŽ‘βŽ’βŽ’βŽ£π΅+π‘Ÿπ΄π΅βˆ—0⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦βˆ’2π‘Ÿπ΄π΅πΆ0.(2.39) Let 𝑃1,𝑃2,𝑃3,𝑃4ξ€»π‘Ž=𝑃,𝑖=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πœ™ξ€·π΄π‘–ξ€Έξ€·π΄0000πœ™π‘–ξ€Έξ€·π΄0000πœ™π‘–ξ€Έ0𝐴000πœ™π‘–ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘,𝑖=1,3,1=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πœ™ξ€·π΅1𝐡0000πœ™1𝐡0000πœ™1ξ€Έ0𝐡000πœ™1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(2.40) Note that πœ™(𝑋0) is a particular solution to (2.13), it is not difficult to find by Lemma 2.3, block Gaussian elimination, and property (𝑒) of πœ™(β‹…) that π‘Ÿξ€Ίπ΄,πΆβˆ—ξ€»βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£=π‘Ÿπ΄π‘ƒ0𝑏𝑇10π‘Ž1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€Ίπœ™ξ€·π΄βˆ’4π‘Ÿ1[]βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1ξ€Έξ€»βˆ’4π‘Ÿπœ™(𝐹)=π‘Ÿ0𝑃4πœ™π‘‡ξ€·πΆ2𝑃𝑇1𝑏𝑇1βˆ’14πœ™ξ€·πΆ1𝑃𝑇1π‘Ž1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£πœ™ξ€·π΅βˆ’4π‘Ÿβˆ—1ξ€Έπœ™ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣0𝑃=π‘Ÿ1ξ€»πœ™,0,0,0𝑇𝐢2𝑃𝑇1𝑏𝑇1πœ™ξ€·πΆ1𝑃𝑇1π‘Ž1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£πœ™ξ€·π΅βˆ’4π‘Ÿβˆ—1ξ€Έπœ™ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎣𝐿=π‘Ÿ21𝑁𝑇1𝑖𝐿11𝑀1π‘–βŽ€βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£πœ™ξ€·π΅βˆ’4π‘Ÿβˆ—1ξ€Έπœ™ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£πœ™ξ€·π΅+3π‘Ÿβˆ—1ξ€Έπœ™ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎣𝐿+𝑛=π‘Ÿ21𝑁𝑇1𝑖𝐿11𝑀1π‘–βŽ€βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£π΅βˆ’4π‘Ÿβˆ—1𝐴1⎀βŽ₯βŽ₯⎦+𝑛.(2.41) Note that 𝐿𝐴=π‘…π΄βˆ—, then πΏπœ™(𝐴)=π‘…πœ™βˆ—(𝐴); henceπ‘ŸβŽ‘βŽ’βŽ’βŽ£π΅π΄π΅βˆ—0⎀βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘ƒ=π‘Ÿπ΄π‘ƒ000𝑇0π‘Žπ‘‡3𝑏1π‘Žπ‘‡10π‘Ž30000𝑏𝑇10000π‘Ž1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦[][]ξ€Ίπœ™ξ€·π΄000βˆ’8π‘Ÿπœ™(𝑀)βˆ’8π‘Ÿπœ™(𝐹)βˆ’8π‘Ÿ1βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£ξ€Έξ€»=π‘Ÿ0𝑀𝑇31𝑁11𝑀𝑇11𝑀31πœ™ξ€·πΆ3ξ€Έπœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1𝑁𝑇11πœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝑀11πœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πœ™ξ€·π΄βˆ’8π‘Ÿ3ξ€Έπœ™ξ€·π΅βˆ—1ξ€Έπœ™ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πœ™ξ€·π΄+6π‘Ÿ3ξ€Έπœ™ξ€·π΅βˆ—1ξ€Έπœ™ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎒⎒⎣+2𝑛=π‘Ÿ0𝑀𝑇31𝑁11𝑀𝑇11𝑀31πœ™ξ€·πΆ3ξ€Έπœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1𝑁𝑇11πœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝑀11πœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄βˆ’8π‘Ÿ3π΅βˆ—1𝐴1⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+2𝑛.(2.42) Similarly, we can obtain π‘ŸβŽ‘βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎒⎒⎣𝐴𝐡𝐢0=π‘Ÿ0𝑁11𝑀𝑇11𝑀31πœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1𝑁𝑇11πœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝑀11πœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄βˆ’4π‘Ÿ3π΅βˆ—1𝐴1⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£π΅βˆ’4π‘Ÿβˆ—1𝐴1⎀βŽ₯βŽ₯⎦+2𝑛,(2.43) Substituting (2.41) and (2.43) into (2.38) and (2.39) yields (2.33) and (2.34), that is 𝑖=1.

Corollary 2.8. Suppose system (1.5) over ℍ have a Hermitian solution. Then we have the following.
(a) System (1.5) has a real hermtian solution if and only if ⎑⎒⎒⎣𝐿2π‘Ÿ21𝑁𝑇1𝑖𝐿11𝑀1π‘–βŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎒⎒⎣+π‘Ÿ0𝑀𝑇31𝑁11𝑀𝑇11𝑀31πœ™ξ€·πΆ3ξ€Έπœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1𝑁𝑇11πœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝑀11πœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎒⎒⎣=2π‘Ÿ0𝑁1𝑖𝑀𝑇1𝑖𝑀31πœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1𝑁𝑇11πœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝑀11πœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦(2.44) hold when 𝑖=2,3,4. In that case, the real solution of (1.5) can be expressed as 𝑋=𝑋1 in (2.15).
(b) System (1.5) has a complex solution if and only if (2.44) hold when 𝑖=3,4 or 𝑖=2,4 or 𝑖=2,3. In that case, the complex solutions of (1.5) can be expressed as 𝑋=𝑋1+𝑋2𝑖 or 𝑋=𝑋1+𝑋3𝑗 or 𝑋=𝑋1+𝑋4π‘˜, where 𝑋1,𝑋2,𝑋3, and 𝑋4 are expressed as (2.15), (2.16), (2.17), and (2.18), respectively.

Proof. From (2.34) we can get the necessary and sufficient conditions for 𝑋𝑖=0,𝑖=1,2,3,4. Thus we can get the results of this Corollary.

3. Solvability Conditions for Real and Complex Hermitian Solutions to (1.6) Over ℍ

In this section, using the results of Theorem 2.6, Theorem 2.7, and Corollary 2.8, we give necessary and sufficient conditions for (1.6) over ℍ to have real and complex Hermitian solutions.

Theorem 3.1. Let 𝐴1,𝐴3,𝐡1,𝐢1,𝐢2, and 𝐢3 be defined in Lemma 2.2, 𝐴4βˆˆβ„π‘™Γ—π‘›,𝐢4βˆˆβ„π‘™Γ—π‘™, and suppose that system (1.5) and the matrix equation 𝐴4π‘Œπ΄βˆ—4=𝐢4 over ℍ have Hermitian solutions 𝑋 and π‘Œβˆˆβ„π‘›Γ—π‘›, respectively. Then system (1.6) over ℍ has a real Hermitian solution if and only if (2.44) hold when 𝑖=2,3,4, and π‘ŸβŽ‘βŽ’βŽ’βŽ£0𝑀𝑇31𝑀31πœ™ξ€·πΆ3ξ€ΈβŽ€βŽ₯βŽ₯βŽ¦ξ€·π‘€=2π‘Ÿ31ξ€Έπ‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£,(3.1)0𝑀𝑇41000𝑀𝑇411πœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·π΄1𝑀41πœ™ξ€·πΆ4ξ€Έπœ™ξ€·π΄4ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄4ξ€Έπœ™π‘‡ξ€·πΆ1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€·π‘€=π‘Ÿ41ξ€ΈβŽ‘βŽ’βŽ’βŽ£π‘€+π‘Ÿπ‘‡41𝑀00𝑇411πœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯⎦,(3.2)π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£00𝑀𝑇41𝑀41𝑀411πœ™ξ€·πΆ4ξ€Έ0πœ™π‘‡ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄4𝐴0πœ™1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄4ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ξ€·π‘€=π‘Ÿ41ξ€ΈβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘€+π‘Ÿ41𝑀4110πœ™π‘‡ξ€·π΅1𝐴0πœ™1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£00𝑀𝑇4100000𝑀𝑇411πœ™π‘‡ξ€·π΄3ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·π΄1𝑀41𝑀411πœ™ξ€·πΆ4𝐴0000πœ™3𝐢0πœ™3ξ€Έπœ™ξ€·π΄3ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ1ξ€Έ0πœ™π‘‡ξ€·π΅1ξ€Έ0πœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΄1𝐴0πœ™1𝐢0πœ™1ξ€Έπœ™π‘‡ξ€·π΄3ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘€=2π‘Ÿ41𝑀411𝐴0πœ™3ξ€Έ0πœ™π‘‡ξ€·π΅1𝐴0πœ™1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£00𝑀𝑇410000𝑀𝑇411πœ™ξ€·π΅1ξ€Έπœ™π‘‡ξ€·π΄1𝑀41𝑀411πœ™ξ€·πΆ4ξ€Έ000πœ™π‘‡ξ€·π΅1ξ€Έ0πœ™π‘‡ξ€·π΅1ξ€Έπœ™ξ€·πΆ2ξ€Έπœ™π‘‡ξ€·π΅1ξ€Έπœ™π‘‡ξ€·πΆ1𝐴0πœ™1𝐢0πœ™1ξ€Έπœ™ξ€·π΅1ξ€Έπœ™ξ€·πΆ1ξ€Έπœ™π‘‡ξ€·π΄1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘€=2π‘Ÿ41𝑀4110πœ™π‘‡ξ€·π΅1𝐴0πœ™1ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,(3.3) where 𝑀41=⎑⎒⎒⎒⎒⎒⎒⎣𝐴42𝐴43𝐴44𝐴41𝐴44βˆ’π΄43βˆ’π΄44𝐴41𝐴42𝐴43βˆ’π΄42𝐴41⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑀411=⎑⎒⎒⎒⎒⎒⎒⎣𝐴21000βˆ’π΄22000βˆ’π΄23000βˆ’π΄24⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦000.(3.4)

Proof. From Corollary 2.8, system (1.5) over ℍ has a real Hermitian solution if and only if (2.44) hold when 𝑖=2,3,4. By (2.15), the real Hermitian solutions of (1.5) over ℍ can be expressed as 𝑋1=14𝑃1πœ™ξ€·π‘‹0𝑃𝑇1+14𝑃2πœ™ξ€·π‘‹0𝑃𝑇2+14𝑃3πœ™ξ€·π‘‹0𝑃𝑇3+14𝑃4πœ™ξ€·π‘‹0𝑃𝑇4+𝑃1,𝑃2,𝑃3,𝑃4ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+𝑃1,𝑃2,𝑃3,𝑃4ξ€»πΏπœ™(𝐴1)πΏπœ™(𝐹)π‘‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(3.5) where 𝑉 is arbitrary matrices with compatible sizes.
Let 𝐴1,𝐢1=0;𝐡1,𝐢2=0;𝐴3=𝐴4; 𝐢3=𝐢4 in Corollary 2.8 and (2.15). It is easy to verify that the matrix equation 𝐴4π‘Œπ΄βˆ—4=𝐢4 over ℍ has a real Hermitian solution if and only if (3.1) hold and the real Hermitian solution can be expressed as π‘Œ1=14𝑃1πœ™ξ€·π‘Œ0𝑃𝑇1+14𝑃2πœ™ξ€·π‘Œ0𝑃𝑇2+14𝑃3πœ™ξ€·π‘Œ0𝑃𝑇3+14𝑃4πœ™ξ€·π‘Œ0𝑃𝑇4+𝑃1,𝑃2,𝑃3,𝑃4ξ€»πΏπœ™(𝐴4)π‘ˆ+π‘ˆπ‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(3.6) where πœ™(π‘Œ0) is a particular solution to πœ™(𝐴4)(π‘Œπ‘–π‘—)4Γ—4πœ™π‘‡(𝐴4)=πœ™(𝐢4) and π‘ˆ is arbitrary matrices with compatible sizes. The expression of π‘Œ1 can also be obtained from Lemma 2.1. Let 𝑃1,𝑃2,𝑃3,𝑃4ξ€»1=𝑃,𝐺=4𝑃1πœ™ξ€·π‘‹0𝑃𝑇1+14𝑃2πœ™ξ€·π‘‹0𝑃𝑇2+14𝑃3πœ™ξ€·π‘‹0𝑃𝑇3+14𝑃4πœ™ξ€·π‘‹0𝑃𝑇4βˆ’14𝑃1πœ™ξ€·π‘Œ0𝑃𝑇1βˆ’14𝑃2πœ™ξ€·π‘Œ0𝑃𝑇2βˆ’14𝑃3πœ™ξ€·π‘Œ0𝑃𝑇3βˆ’14𝑃4πœ™ξ€·π‘Œ0𝑃𝑇4.(3.7) Equating 𝑋1 and π‘Œ1, we obtain the following equation: 𝑋1βˆ’π‘Œ1=𝐺+π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)π‘‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’π‘ƒπΏπœ™(𝐴4)π‘ˆβˆ’π‘ˆπ‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐴4)𝑃𝑇1πΏπœ™(𝐴4)𝑃𝑇2πΏπœ™(𝐴4)𝑃𝑇3πΏπœ™(𝐴4)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.8) It is obvious that system (1.5) and the matrix equation 𝐴4π‘Œπ΄βˆ—4=𝐢4 over ℍ have common real Hermitian solution if and only if minπ‘Ÿ(𝑋1βˆ’π‘Œ1)=0,thatis,𝑋1βˆ’π‘Œ1=0. Hence, we have the matrix equation 𝐺=π‘ƒπΏπœ™(𝐴4)π‘ˆ+π‘ˆπ‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐴4)𝑃𝑇1πΏπœ™(𝐴4)𝑃𝑇2πΏπœ™(𝐴4)𝑃𝑇3πΏπœ™(𝐴4)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)π‘‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇1πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇2πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇3πΏπœ™(𝑀)πΏπœ™(𝐹)πΏπœ™(𝐴1)𝑃𝑇4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(3.9) We know by Lemma 2.5 that (3.9) is solvable if and only if the following four rank equalities hold π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£πΊπ‘ƒπΏπœ™(𝐴4)π‘…πœ™(𝐴4)𝑃𝑇0π‘…πœ™(𝐹)π‘…πœ™(𝐴1)𝑃𝑇0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣=π‘Ÿ0π‘ƒπΏπœ™(𝐴4)π‘…πœ™(𝐴4)𝑃𝑇0π‘…πœ™(𝐹)π‘…πœ™(𝐴1)𝑃𝑇0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,π‘ŸβŽ‘βŽ’βŽ’βŽ£πΊπ‘ƒπΏπœ™(𝐴4)π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)π‘…πœ™(𝐴4)π‘ƒπ‘‡βŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎣00=π‘Ÿ0π‘ƒπΏπœ™(𝐴4)π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)π‘…πœ™(𝐴4)π‘ƒπ‘‡βŽ€βŽ₯βŽ₯⎦,π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£00πΊπ‘ƒπΏπœ™(𝐴4)π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘…πœ™(𝐴4)𝑃𝑇𝑅00πœ™(𝑀)π‘…πœ™(𝐹)π‘…πœ™(𝐴1)π‘ƒπ‘‡βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣00=π‘Ÿ0π‘ƒπΏπœ™(𝐴4)π‘ƒπΏπœ™(𝐴1)πΏπœ™(𝐹)πΏπœ™(𝑀)π‘…πœ™(𝐴4)𝑃𝑇𝑅00πœ™(𝑀)π‘…πœ™(𝐹)π‘…πœ™(𝐴1)π‘ƒπ‘‡βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£00𝐺𝑃𝐿