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

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=𝐶₁,𝑋𝐵₁=𝐶₂,and𝐴₃𝑋𝐴^∗3=𝐶₃. Moreover, formulas of the maximal and minimal ranks of four real matrices 𝑋₁,𝑋₂,𝑋₃, and 𝑋₄ in solution 𝑋=𝑋₁+𝑋₂𝑖+𝑋₃𝑗+𝑋₄𝑘 to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations 𝐴₁𝑋=𝐶₁,𝑋𝐵₁=𝐶₂,𝐴₃𝑋𝐴^∗3=𝐶₃, and𝐴₄𝑋𝐴^∗4=𝐶₄ to have real and complex Hermitian solutions.

1. Introduction

Throughout this paper, we denote the real number field by ℝ; the complex field by ℂ; the set of all 𝑚×𝑛 matrices over the quaternion algebra ℍ=𝑎₀+𝑎₁𝑖+𝑎₂𝑗+𝑎₃𝑘∣𝑖²=𝑗²=𝑘²=𝑖𝑗𝑘=−1,𝑎₀,𝑎₁,𝑎₂,𝑎₃∈ℝ(1.1) by ℍ^𝑚×𝑛; 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 𝐴^𝑇,𝐴^∗,ℛ(𝐴), 𝒩(𝐴), respectively; the dimension of ℛ(𝐴) by dimℛ(𝐴). By [1], for a quaternion matrix 𝐴,dimℛ(𝐴)=dim𝒩(𝐴). dimℛ(𝐴) is called the rank of a quaternion matrix 𝐴 and denoted by 𝑟(𝐴). The Moore-Penrose inverse of matrix 𝐴 over ℍ by 𝐴^† which satisfies four Penrose equations 𝐴𝐴^†𝐴=𝐴,𝐴^†𝐴𝐴^†=𝐴^†,(𝐴𝐴^†)^∗=𝐴𝐴^†, and (𝐴^†𝐴)^∗=𝐴^†𝐴. In this case, 𝐴^† is unique and (𝐴^†)^∗=(𝐴^∗)^†. Moreover, 𝑅_𝐴 and 𝐿_𝐴 stand for the two projectors 𝐿_𝐴=𝐼−𝐴^†𝐴, and 𝑅_𝐴=𝐼−𝐴𝐴^† induced by 𝐴. Clearly, 𝑅_𝐴 and 𝐿_𝐴 are idempotent and satisfies (𝑅_𝐴)^∗=𝑅_𝐴,(𝐿_𝐴)^∗=𝐿_𝐴,𝑅_𝐴=𝐿_𝐴^∗, and 𝑅_𝐴^∗=𝐿_𝐴.

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=𝐵,𝐴𝑋𝐵=𝐶 and 𝐴₁𝑋=𝐶₁,𝑋𝐵₂=𝐶₂,(1.2) over the complex field ℂ, 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 𝐴𝑋𝐴^∗=𝐵(1.3) over ℂ in terms of generalized inverses, respectively. In [10], Tian and Liu established the solvability conditions for 𝐴₃𝑋𝐴^∗3=𝐶₃,𝐴₄𝑋𝐴^∗4=𝐶₄(1.4) to have a common Hermitian solution over ℂ 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 𝐴𝐴₁𝑋=𝐶₁,𝑋𝐵₂=𝐶₂,𝐴₃𝑋𝐴^∗3=𝐶₃,(1.5)₁𝑋=𝐶₁,𝑋𝐵₂=𝐶₂,𝐴₃𝑋𝐴^∗3=𝐶₃,𝐴₄𝑋𝐴^∗4=𝐶₄,(1.6) for operators between Hilbert 𝐶^∗-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 𝐴𝑋𝐵=𝐶 over ℂ 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 𝐵^∗=𝐵. 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 𝐴𝑋𝐵=𝐶, 𝐴₁𝑋𝐵𝐴₁𝑋=𝐴𝐶_1∗,𝐴₂𝑋𝐵₂=𝐶₂,+𝐵𝑋𝐵^∗=𝐶,(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+𝑋₄𝑘 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+𝑋₄𝑘 to (1.5) over ℍ, last, investigate the real and complex Hermitian solutions to (1.5) over ℍ.

For an arbitrary matrix 𝑀_𝑡=𝑀_𝑡1+𝑀_𝑡2𝑖+𝑀_𝑡3𝑗+𝑀_𝑡4𝑘∈ℍ^𝑚×𝑛 where 𝑀_𝑡1,𝑀_𝑡2,𝑀_𝑡3, and 𝑀_𝑡4 are real matrices, we define a map 𝜙(⋅) from ℍ^𝑚×𝑛 to ℝ^4𝑚×4𝑛 by 𝜙𝑀_𝑡=⎡⎢⎢⎢⎢⎢⎢⎣𝑀^𝑡1𝑀_𝑡2𝑀_𝑡3𝑀_𝑡4−𝑀_𝑡2𝑀_𝑡1𝑀_𝑡4−𝑀_𝑡3−𝑀_𝑡3−𝑀_𝑡4𝑀_𝑡1𝑀_𝑡2−𝑀_𝑡4𝑀_𝑡3−𝑀_𝑡2𝑀_𝑡1⎤⎥⎥⎥⎥⎥⎥⎦.(2.1) By (2.1), it is easy to verify that 𝜙(⋅) satisfies the following properties.(a)𝑀=𝑁⇔𝜙(𝑀)=𝜙(𝑁). (b)𝜙(𝑘𝑀+𝑙𝑁)=𝑘𝜙(𝑀)+𝑙𝜙(𝑁),𝜙(𝑀𝑁)=𝜙(𝑀)𝜙(𝑁),𝑘,𝑙∈ℝ.(c)𝜙(𝑀^∗)=𝜙^𝑇(𝑀), 𝜙(𝑀^†)=𝜙^†(𝑀). (d)𝜙(𝑀)=𝑇^𝑚−1𝜙(𝑀)𝑇_𝑛=𝑅^𝑚−1𝜙(𝑀)𝑅_𝑛=𝑆^𝑚−1𝜙(𝑀)𝑆_𝑛, where 𝑡=𝑚,𝑛, 𝑇_𝑡=⎡⎢⎢⎢⎢⎢⎢⎣0−𝐼^𝑡𝐼00_𝑡00000𝐼0_𝑡00−𝐼_𝑡0⎤⎥⎥⎥⎥⎥⎥⎦,𝑅^𝑡=⎡⎢⎢⎢⎢⎢⎢⎣00−𝐼^𝑡000−0𝐼_𝑡𝐼0_𝑡𝐼0_𝑡0000⎤⎥⎥⎥⎥⎥⎥⎦,𝑆^𝑡=⎡⎢⎢⎢⎢⎢⎢⎣000−𝐼^𝑡00−0𝐼_𝑡𝐼_𝑡0𝐼00_𝑡000⎤⎥⎥⎥⎥⎥⎥⎦.(2.2)(e)𝑟[𝜙(𝑀)]=4𝑟(𝑀). (f)𝑀^∗=𝑀⇔𝜙^𝑇(𝑀)=𝜙(𝑀), 𝑀^∗=−𝑀⇔𝜙^𝑇(𝑀)=−𝜙(𝑀).

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

Lemma 2.1 (see [2, Lemma  2.1]). Let 𝐴∈ℍ^𝑚×𝑛,𝐵=𝐵^∗∈ℍ^𝑚×𝑚 be known, 𝑋∈ℍ^𝑛×𝑛 unknown; then the system (1.3) has a Hermitian solution if and only if 𝐴𝐴^†𝐵=𝐵.(2.3) In that case, the general Hermitian solution of (1.3) can be expressed as 𝑋=𝐴^†𝐵𝐴^†^∗+𝐿_𝐴𝑉+𝑉^∗𝐿_𝐴,(2.4) where 𝑉 is arbitrary matrix over ℍ with compatible size.

Lemma 2.2 (see [12, Corollary  3.4]). Let 𝐴₁,𝐶₁∈ℍ^𝑚×𝑛;𝐵₁,𝐶₂∈ℍ^𝑛×𝑠;𝐴₃∈ℍ^𝑟×𝑛;𝐶₃∈ℍ^𝑟×𝑟 be known, 𝑋∈ℍ^𝑛×𝑛 unknown, and 𝐹=𝐵^∗1𝐿_𝐴1,𝑀=𝑆𝐿_𝐹,𝑆=𝐴₃𝐿_𝐴1,𝐷=𝐶^∗2−𝐵^∗1𝐴^†1𝐶₁,𝐽=𝐴^†1𝐶₁+𝐹^†𝐷,𝐺=𝐶₃−𝐴₃(𝐽+𝐿_𝐴1𝐿^∗𝐹𝐽^∗)𝐴^∗3,𝐶₃=𝐶^∗3; then the system (1.5) have a Hermitian solution if and only if 𝐴₁𝐶₂𝑅=𝐶_1𝐴1𝐵₁,𝐴₁𝐶^∗1=𝐶₁𝐴^∗1,𝐵^∗1𝐶₂=𝐶^∗2𝐵₁,𝐶₁=0,𝑅_𝐹𝐷=0,𝑅_𝑀𝐺=0.(2.5) In that case, the general Hermitian solution of (1.5) can be expressed as 𝑋=𝐽+𝐿_𝐴1𝐿_𝐹𝐽^∗+𝐿_𝐴1𝐿_𝐹𝑀^†𝐺𝑀^†^∗𝐿_𝐹𝐿_𝐴1+𝐿_𝐴1𝐿_𝐹𝐿_𝑀𝑉𝐿_𝐹𝐿_𝐴1+𝐿_𝐴1𝐿_𝐹𝑉^∗𝐿_𝑀𝐿_𝐹𝐿_𝐴1,(2.6) where 𝑉is arbitrary matrix over ℍ with compatible size.

Lemma 2.3 (see [21, Lemma  2.4]). Let 𝐴∈ℍ^𝑚×𝑛,𝐵∈ℍ^𝑚×𝑘,𝐶∈ℍ^𝑙×𝑛,𝐷∈ℍ^𝑗×𝑘, and 𝐸∈ℍ^𝑙×𝑖. Then they satisfy the following rank equalities.(a)𝑟(𝐶𝐿_𝐴)=𝑟^𝐴𝐶−𝑟(𝐴). (b)𝑟[^𝐵𝐴𝐿_𝐶]=𝑟^{𝐵𝐴0𝐶}−𝑟(𝐶). (c)𝑟^𝐶𝑅_𝐵𝐴=𝑟^{𝐶0𝐴𝐵}−𝑟(𝐵). (d)𝑟^{𝐴𝐵𝐿𝐷}_{𝑅𝐸𝐶0}=𝑟^{𝐴𝐵0𝐶0𝐸0𝐷0}−𝑟(𝐷)−𝑟(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=±𝐴^∗∈ℍ^𝑚×𝑚,𝐵∈ℍ^𝑚×𝑛, and𝐶∈ℍ^𝑝×𝑚 be given; then max_{𝑋∈ℍ𝑛×𝑝}𝑟𝐴−𝐵𝑋𝐶∓(𝐵𝑋𝐶)^∗=min⎧⎪⎨⎪⎩𝑟𝐴𝐵𝐶^∗,𝑟⎡⎢⎢⎣𝐵𝐴𝐵_∗min_{𝑋∈ℍ𝑛×𝑝}0⎤⎥⎥⎦,𝑟⎡⎢⎢⎣𝐶𝐴𝐶0^∗⎤⎥⎥⎦⎫⎪⎬⎪⎭,𝑟𝐴−𝐵𝑋𝐶∓(𝐵𝑋𝐶)^∗=2𝑟𝐴𝐵𝐶^∗+max𝑠¹,𝑠₂,(2.7) where 𝑠₁=𝑟⎡⎢⎢⎣𝐵𝐴𝐵_∗𝑠0⎤⎥⎥⎦−2𝑟⎡⎢⎢⎣𝐵𝐴^∗𝐵0𝐶0^∗₂⎤⎥⎥⎦,=𝑟⎡⎢⎢⎣𝐴𝐶^∗𝐶0⎤⎥⎥⎦−2𝑟⎡⎢⎢⎣𝐶𝐴𝐵0𝐶0^∗⎤⎥⎥⎦.(2.8) If ℛ(𝐵)⊆ℛ(𝐶^∗), max_{𝑋∈ℍ𝑛×𝑝}𝑟𝐴−𝐵𝑋𝐶−(𝐵𝑋𝐶)^∗=min⎧⎪⎨⎪⎩𝑟𝐴𝐶^∗,𝑟⎡⎢⎢⎣𝐵𝐴𝐵_∗min_{𝑋∈ℍ𝑛×𝑝}0⎤⎥⎥⎦⎫⎪⎬⎪⎭,(2.9)𝑟𝐴−𝐵𝑋𝐶−(𝐵𝑋𝐶)^∗=2𝑟𝐴𝐶^∗+𝑟⎡⎢⎢⎣𝐵𝐴𝐵_∗0⎤⎥⎥⎦−2𝑟⎡⎢⎢⎣𝐶𝐴𝐵0⎤⎥⎥⎦.(2.10)

Lemma 2.5 (see [28, Theorem  3.1]). Let 𝐴∈ℍ^𝑚×𝑛,𝐵₁∈ℍ^𝑚×𝑝1,𝐵₃∈ℍ^𝑚×𝑝3,𝐵₄∈ℍ^𝑚×𝑝4,𝐶₂∈ℍ^𝑞2×𝑛,𝐶₃∈ℍ^𝑞3×𝑛, and 𝐶₄∈ℍ^𝑞4×𝑛 be given. Then the matrix equation 𝐵₁𝑋₁+𝑋₂𝐶₂+𝐵₃𝑋₃𝐶₃+𝐵₄𝑋₄𝐶₄=𝐴(2.11) is consistent if and only if 𝑟⎡⎢⎢⎢⎢⎢⎢⎣𝐴𝐵¹𝐶𝐶₂₃0𝐶₄00⎤⎥⎥⎥⎥⎥⎥⎦=𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝐵¹𝐶𝐶₂₃0𝐶₄00⎤⎥⎥⎥⎥⎥⎥⎦,𝑟⎡⎢⎢⎣𝐴𝐵¹𝐵₃𝐵₄𝐶₂000⎤⎥⎥⎦=𝑟⎡⎢⎢⎣0𝐵¹𝐵₃𝐵₄𝑟⎡⎢⎢⎢⎢⎣𝐴𝐵₁𝐶₂000⎤⎥⎥⎦,𝐵₃𝐶𝐶₂00₄00⎤⎥⎥⎥⎥⎦=𝑟⎡⎢⎢⎢⎢⎣0𝐵¹𝐵₃𝐶𝐶₂00₄00⎤⎥⎥⎥⎥⎦,𝑟⎡⎢⎢⎢⎢⎣𝐴𝐵¹𝐵₄𝐶𝐶₂00₃00⎤⎥⎥⎥⎥⎦=𝑟⎡⎢⎢⎢⎢⎣0𝐵¹𝐵₄𝐶𝐶₂00₃00⎤⎥⎥⎥⎥⎦.(2.12)

Theorem 2.6. System (1.5) has a Hermitian solution over ℍ if and only if the system of matrix equations 𝜙𝐴₁𝑌_𝑖𝑗_4×4=𝜙𝐶₁,𝑌_𝑖𝑗_4×4𝜙𝐵₁=𝜙𝐶₂,𝜙𝐴₃𝑌_𝑖𝑗_4×4𝜙^𝑇𝐴₃𝑖,𝑗==𝜙1,𝐶2,₃3,,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==𝑋1₁+𝑋₂𝑖+𝑋₃𝑗+𝑋₄𝑘4𝑌¹¹+𝑌₂₂+𝑌₃₃+𝑌₄₄+14𝑌¹²+1−𝑌^𝑇12+𝑌₃₄−𝑌^𝑇34𝑖4𝑌¹³−𝑌^𝑇13+𝑌^𝑇24−𝑌₂₄𝑗+14𝑌¹⁴−𝑌^𝑇14+𝑌₂₃−𝑌^𝑇23𝑘,(2.14) where 𝑌_𝑡𝑡=𝑌^𝑇𝑡𝑡;𝑡=1,2,3,4;𝑌^𝑇1𝑗=𝑌_𝑗1;𝑗=2,3,4;𝑌^𝑇2𝑗=𝑌_𝑗2;𝑗=3,4;𝑌^𝑇34=𝑌₄₃ are the general solutions of (2.13) over ℝ. Written in an explicit form, 𝑋₁,𝑋₂,𝑋₃, and 𝑋₄ in (2.14) are 𝑋₁=14𝑃¹𝜙𝑋₀𝑃^𝑇1+14𝑃²𝜙𝑋₀𝑃^𝑇2+14𝑃³𝜙𝑋₀𝑃^𝑇3+14𝑃⁴𝜙𝑋₀𝑃^𝑇4+𝑃₁,𝑃₂,𝑃₃,𝑃₄𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉⎡⎢⎢⎢⎢⎢⎢⎣𝐿^𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4+𝑃₁⎤⎥⎥⎥⎥⎥⎥⎦,𝑃₂,𝑃₃,𝑃₄𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝑉^𝑇⎡⎢⎢⎢⎢⎢⎢⎣𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4𝑋⎤⎥⎥⎥⎥⎥⎥⎦,(2.15)₂=14𝑃¹𝜙𝑋₀𝑃^𝑇2−14𝑃²𝜙𝑋₀𝑃^𝑇1+14𝑃³𝜙𝑋₀𝑃^𝑇4−14𝑃⁴𝜙𝑋₀𝑃^𝑇3+𝑃₁,−𝑃₂,𝑃₃,−𝑃₄𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉⎡⎢⎢⎢⎢⎢⎢⎣𝐿^𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3+𝑃₂⎤⎥⎥⎥⎥⎥⎥⎦,𝑃₁,𝑃₄,𝑃₃𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝑉^𝑇⎡⎢⎢⎢⎢⎢⎢⎣𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1−𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3−𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4𝑋⎤⎥⎥⎥⎥⎥⎥⎦,(2.16)₃=14𝑃¹𝜙𝑋₀𝑃^𝑇3−14𝑃³𝜙𝑋₀𝑃^𝑇1+14𝑃⁴𝜙𝑋₀𝑃^𝑇2−14𝑃²𝜙𝑋₀𝑃^𝑇4+𝑃₁,−𝑃₃,𝑃₄,−𝑃₂𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉⎡⎢⎢⎢⎢⎢⎢⎣𝐿^𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4+𝑃₃⎤⎥⎥⎥⎥⎥⎥⎦,𝑃₁,𝑃₂,𝑃₄𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝑉^𝑇⎡⎢⎢⎢⎢⎢⎢⎣𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1−𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4−𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝑋⎤⎥⎥⎥⎥⎥⎥⎦,(2.17)₄=14𝑃¹𝜙𝑋₀𝑃^𝑇4−14𝑃⁴𝜙𝑋₀𝑃^𝑇1+14𝑃²𝜙𝑋₀𝑃^𝑇3−14𝑃³𝜙𝑋₀𝑃^𝑇2+𝑃₁,−𝑃₄,𝑃₂,−𝑃₃𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉⎡⎢⎢⎢⎢⎢⎢⎣𝐿^𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2+𝑃₄⎤⎥⎥⎥⎥⎥⎥⎦,𝑃₁,𝑃₃,𝑃₂𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝑉^𝑇⎡⎢⎢⎢⎢⎢⎢⎣𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1−𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2−𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3⎤⎥⎥⎥⎥⎥⎥⎦,(2.18) where 𝑃₁=𝐼_𝑛,0,0,0,𝑃₂=0,𝐼_𝑛,0,0,𝑃₃=0,0,𝐼_𝑛,0,𝑃₄=0,0,0,𝐼_𝑛,(2.19)𝜙(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 𝜙(⋅) to (1.5) yields 𝜙𝐴₁𝜙(𝑋)=𝜙𝐶₁,𝜙(𝑋)𝜙𝐵₂=𝜙𝐶₂,𝜙𝐴₃𝜙(𝑋)𝜙^𝑇𝐴₃=𝜙𝐶₃,(2.20) implying that 𝜙(𝑋) is a real symmetric solution to (2.13).
Conversely, suppose that (2.13) has a real symmetric solution 𝑌=𝑌^𝑇=𝑌_𝑖𝑗_4×4,𝑖,𝑗=1,2,3,4.(2.21) That is, 𝜙𝐴₁𝑌=𝜙𝐶₁,𝑌𝜙𝐵₂=𝜙𝐶₂,𝜙𝐴₃𝑌𝜙^𝑇𝐴₃=𝜙𝐶₃,(2.22) then by property (𝑑) of 𝜙(⋅), 𝑇^𝑚−1𝜙𝐴₁𝑇𝑇_𝑛𝑌_𝑟−1=𝑇^𝑚−1𝜙𝐶₁𝑇_𝑛,𝑌𝑇^𝑛−1𝜙𝐵₂𝑇_𝑠=𝑇^𝑛−1𝜙𝐶₂𝑇_𝑠,𝑅_𝑚−1𝜙𝐴₃𝑇_𝑛𝑌𝑇^𝑛−1𝜙^𝑇𝐴₃𝑇_𝑟=𝑇^𝑟−1𝜙𝐶₃𝑇_𝑟,𝜙𝐴₁𝑅𝑅_𝑛𝑌_𝑟−1=𝑅^𝑚−1𝜙𝐶₁𝑅_𝑛,𝑌𝑅^𝑛−1𝜙𝐵₂𝑅_𝑠=𝑅^𝑛−1𝜙𝐶₂𝑅_𝑠,𝑆_𝑚−1𝜙𝐴₃𝑅_𝑛𝑌𝑅_𝑛𝜙^𝑇𝐴₃𝑅_𝑟=𝑅^𝑟−1𝜙𝐶₃𝑅_𝑟,𝜙𝐴₁𝑆𝑆_𝑛𝑌_𝑟−1=𝑆^𝑚−1𝜙𝐶₁𝑆_𝑛,𝑌𝑆^𝑛−1𝜙𝐵₂𝑆_𝑠=𝑆^𝑛−1𝜙𝐶₂𝑆_𝑠,𝜙𝐴₃𝑆_𝑛𝑌𝑆^𝑛−1𝜙^𝑇𝐴₃𝑆_𝑟=𝑆^𝑟−1𝜙𝐶₃𝑆_𝑟.(2.23) Hence, 𝜙𝜙𝐴𝐴₁₁𝑇_𝑛𝑌𝑇^𝑛−1=𝜙𝐶₁,𝑇_𝑛𝑌𝑇^𝑛−1𝜙𝐵₂=𝜙𝐶₂,𝜙𝐴₃𝑇_𝑛𝑌𝑇^𝑛−1𝜙^𝑇𝐴₃=𝜙𝐶₃,𝜙𝐴₁𝑅_𝑛𝑌𝑅^𝑛−1=𝜙𝐶₁,𝑅_𝑛𝑌𝑅^𝑛−1𝜙𝐵₂=𝜙𝐶₂,𝜙𝐴₃𝑅_𝑛𝑌𝑅^𝑛−1𝜙^𝑇𝐴₃=𝜙𝐶₃,𝑆_𝑛𝑌𝑆^𝑛−1=𝜙𝐶₁,𝑆_𝑛𝑌𝑆^𝑛−1𝜙𝐵₂=𝜙𝐶₂,𝜙𝐴₃𝑆_𝑛𝑌𝑆^𝑛−1𝜙^𝑇𝐴₃=𝜙𝐶₃,(2.24) implying that 𝑇_𝑛𝑌𝑇^𝑛−1,𝑅_𝑛𝑌𝑅^𝑛−1, and 𝑆_𝑛𝑌𝑆^𝑛−1 are also symmetric solutions of (2.13). Thus, 14𝑌+𝑇^𝑛𝑌𝑇^𝑛−1+𝑅_𝑛𝑌𝑅^𝑛−1+𝑆_𝑛𝑌𝑆^𝑛−1(2.25) is a symmetric solution of (2.13), where 𝑌+𝑇_𝑛𝑌𝑇^𝑛−1+𝑅_𝑛𝑌𝑅^𝑛−1+𝑆_𝑛𝑌𝑆^𝑛−1=𝑌_𝑖𝑗𝑌_4×4,𝑖=1,2,3,4,₁₁𝑌₁₃=𝑌₁₁+𝑌₂₂+𝑌₃₃+𝑌₄₄,𝑌₁₂=𝑌₁₂−𝑌^𝑇12+𝑌₃₄−𝑌^𝑇34,𝑌₂₁=𝑌₁₃−𝑌^𝑇13+𝑌^𝑇24−𝑌₂₄,𝑌₁₄=𝑌₁₄−𝑌^𝑇14+𝑌₂₃−𝑌^𝑇23,𝑌₂₃=𝑌^𝑇12−𝑌₁₂+𝑌^𝑇34−𝑌₃₄,𝑌₂₂=𝑌₁₁+𝑌₂₂+𝑌₃₃+𝑌₄₄,𝑌₃₁=𝑌₁₄−𝑌^𝑇14+𝑌₂₃−𝑌^𝑇23,𝑌₂₄=𝑌₁₃−𝑌^𝑇13+𝑌^𝑇24−𝑌₂₄,𝑌₃₃=𝑌^𝑇13−𝑌₁₃+𝑌₂₄−𝑌^𝑇24,𝑌₃₂=𝑌₁₄−𝑌^𝑇14+𝑌₂₃−𝑌^𝑇23,𝑌₄₁=𝑌₁₁+𝑌₂₂+𝑌₃₃+𝑌₄₄,𝑌₃₄=𝑌₁₂−𝑌^𝑇12+𝑌₃₄−𝑌^𝑇34,𝑌₄₃=𝑌^𝑇14−𝑌₁₄+𝑌^𝑇23−𝑌₂₃,𝑌₄₂=𝑌₁₃−𝑌^𝑇13+𝑌^𝑇24−𝑌₂₄,=𝑌₁₂−𝑌^𝑇12+𝑌₃₄−𝑌^𝑇34,𝑌₄₄=𝑌₁₁+𝑌₂₂+𝑌₃₃+𝑌₄₄.(2.26) Let 𝑋=14𝑌¹¹+𝑌₂₂+𝑌₃₃+𝑌₄₄+14𝑌¹²+1−𝑌^𝑇12+𝑌₃₄−𝑌^𝑇34𝑖4𝑌¹³−𝑌^𝑇13+𝑌^𝑇24−𝑌₂₄𝑗+14𝑌¹⁴−𝑌^𝑇14+𝑌₂₃−𝑌^𝑇23𝑘.(2.27) Then by (2.1), 𝜙𝑋=14𝑌+𝑇^𝑛𝑌𝑇^𝑛−1+𝑅_𝑛𝑌𝑅^𝑛−1+𝑆_𝑛𝑌𝑆^𝑛−1.(2.28) Hence, by the property (a), we know that 𝑋 is a Hermitian solution of (1.5). Observe that 𝑌_𝑖𝑗,𝑖,𝑗=1,2,3,4, in (2.13) can be written as 𝑌_𝑖𝑗=𝑃_𝑖𝑌𝑃^𝑇𝑗.(2.29) From Lemma 2.2, the general Hermitian solution to (2.13) can be written as 𝑌=𝜙𝑋₀+4𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)+4𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑉^𝑇𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1),(2.30) where 𝑉∈ℝ is arbitrary. Hence, 𝑌_𝑖𝑗=𝑃_𝑖𝜙𝑋₀𝑃^𝑇𝑗+4𝑃_𝑖𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝑃^𝑇𝑗+4𝑃_𝑖𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑉^𝑇𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇𝑗,(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+𝑋₄𝑘 to (1.5) over ℍ.

Theorem 2.7. Suppose that system (1.5) over ℍ has a Hermitian solution, and 𝐴₁=𝐴₁₁+𝐴₁₂𝑖+𝐴₁₃𝑗+𝐴₁₄𝑘,𝐶₁=𝐶₁₁+𝐶₁₂𝑖+𝐶₁₃𝑗+𝐶₁₄𝑘∈ℍ^𝑚×𝑛𝐵ℍ₁=𝐵₁₁+𝐵₁₂𝑖+𝐵₁₃𝑗+𝐵₁₄𝑘,𝐶₂=𝐶₂₁+𝐶₂₂𝑖+𝐶₂₃𝑗+𝐶₂₄𝑘∈ℍ^𝑛×𝑠,𝐴₃=𝐴₃₁+𝐴₃₂𝑖+𝐴₃₃𝑗+𝐴₃₄𝑘∈_𝑟×𝑛,𝐶₃=𝐶₃₁+𝐶₃₂𝑖+𝐶₃₃𝑗+𝐶₃₄𝑘∈ℍ^𝑟×𝑟𝑆₁=⎧⎪⎨⎪⎩𝑋₁∈ℝ^𝑛×𝑛∣𝐴¹𝑋=𝐶₁,𝑋𝐵₁=𝐶₂,𝐴₃𝑋𝐴^∗3=𝐶₃𝑆₂𝑋=𝑋₁+𝑋₂𝑖+𝑋₃𝑗+𝑋₄𝑘⎫⎪⎬⎪⎭,=⎧⎪⎨⎪⎩𝑋₂∈ℝ^𝑝×𝑞∣𝐴¹𝑋=𝐶₁,𝑋𝐵₁=𝐶₂,𝐴₃𝑋𝐴^∗3=𝐶₃𝑆₃𝑋=𝑋₁+𝑋₂𝑖+𝑋₃𝑗+𝑋₄𝑘⎫⎪⎬⎪⎭,=⎧⎪⎨⎪⎩𝑋₃∈ℝ^𝑝×𝑞∣𝐴¹𝑋=𝐶₁,𝑋𝐵₁=𝐶₂,𝐴₃𝑋𝐴^∗3=𝐶₃𝑆₄𝑋=𝑋₁+𝑋₂𝑖+𝑋₃𝑗+𝑋₄𝑘⎫⎪⎬⎪⎭,=⎧⎪⎨⎪⎩𝑋₄∈ℝ^𝑝×𝑞∣𝐴¹𝑋=𝐶₁,𝑋𝐵₁=𝐶₂,𝐴₃𝑋𝐴^∗3=𝐶₃𝐿₂₁𝑋=𝑋₁+𝑋₂𝑖+𝑋₃𝑗+𝑋₄𝑘⎫⎪⎬⎪⎭,=⎡⎢⎢⎢⎢⎢⎢⎣𝐶²¹𝐶₂₂𝐶₂₃𝐶₂₄⎤⎥⎥⎥⎥⎥⎥⎦,𝐿₁₁=⎡⎢⎢⎢⎢⎢⎢⎣𝐶¹¹−𝐶₁₂−𝐶₁₃−𝐶₁₄⎤⎥⎥⎥⎥⎥⎥⎦,𝑀₃₁=⎡⎢⎢⎢⎢⎢⎢⎣𝐴³²𝐴₃₃𝐴₃₄𝐴₃₁𝐴₃₄−𝐴₃₃−𝐴₃₄𝐴₃₁𝐴₃₂𝐴₃₃−𝐴₃₂𝐴₃₁𝑀₁₁⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣𝐴¹²𝐴₁₃𝐴₁₄𝐴₁₁𝐴₁₄−𝐴₁₃−𝐴₁₄𝐴₁₁𝐴₁₂𝐴₁₃−𝐴₁₂𝐴₁₁⎤⎥⎥⎥⎥⎥⎥⎦,𝑀₁₂=⎡⎢⎢⎢⎢⎢⎢⎣𝐴¹¹𝐴₁₃𝐴₁₄−𝐴₁₂𝐴₁₄−𝐴₁₃−𝐴₁₃𝐴₁₁𝐴₁₂−𝐴₁₄−𝐴₁₂𝐴₁₁𝑀₁₃⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣𝐴¹¹𝐴₁₂𝐴₁₄−𝐴₁₂𝐴₁₁−𝐴₁₃−𝐴₁₃−𝐴₁₄𝐴₁₂−𝐴₁₄𝐴₁₃𝐴₁₁⎤⎥⎥⎥⎥⎥⎥⎦,𝑀₁₄=⎡⎢⎢⎢⎢⎢⎢⎣𝐴¹¹𝐴₁₂𝐴₁₃−𝐴₁₂𝐴₁₁𝐴₁₄−𝐴₁₃−𝐴₁₄𝐴₁₁−𝐴₁₄𝐴₁₃−𝐴₁₂𝑁₁₁⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎣−𝐵¹²𝐵₁₁𝐵₁₄−𝐵₁₃−𝐵₁₃−𝐵₁₄𝐵₁₁𝐵₁₂−𝐵₁₄𝐵₁₃−𝐵₁₂𝐵₁₁⎤⎥⎥⎥⎥⎦,𝑁₁₂=⎡⎢⎢⎢⎢⎣𝐵¹¹𝐵₁₂𝐵₁₃𝐵₁₄−𝐵₁₃−𝐵₁₄𝐵₁₁𝐵₁₂−𝐵₁₄𝐵₁₃−𝐵₁₂𝐵₁₁𝑁₁₃⎤⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎣𝐵¹¹𝐵₁₂𝐵₁₃𝐵₁₄−𝐵₁₂𝐵₁₁𝐵₁₄−𝐵₁₃−𝐵₁₄𝐵₁₃−𝐵₁₂𝐵₁₁⎤⎥⎥⎥⎥⎦,𝑁₁₄=⎡⎢⎢⎢⎢⎣𝐵¹¹𝐵₁₂𝐵₁₃𝐵₁₄−𝐵₁₂𝐵₁₁𝐵₁₄−𝐵₁₃−𝐵₁₃−𝐵₁₄𝐵₁₁𝐵₁₂⎤⎥⎥⎥⎥⎦.(2.32) Then the maximal and minimal ranks of 𝑋_𝑖,𝑖=1,2,3,4, in Hermitian solution 𝑋=𝑋₁+𝑋₂𝑖+𝑋₃𝑗+𝑋₄𝑘 to (1.5) are given bymmianx_{𝑋𝑖∈𝑆𝑖𝑋𝑖∈𝑆𝑖}𝑟𝑋_𝑖=min𝑡_1𝑖,𝑡,(2.33)𝑟𝑋_𝑖=2𝑡_1𝑖+𝑡−2𝑡_2𝑖,(2.34) where 𝑡_1𝑖=𝑟⎡⎢⎢⎣𝐿²¹𝑁^𝑇1𝑖𝐿₁₁𝑀_1𝑖⎤⎥⎥⎦−4𝑟⎡⎢⎢⎣𝐵^∗1𝐴₁𝑡=𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝑀_𝑇31⎤⎥⎥⎦+𝑛,𝑁₁₁𝑀^𝑇11𝑀𝑁_31𝑇11𝜙𝐶₃𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁𝑀₁₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₃𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁𝜙𝐶₁𝜙^𝑇𝐴₃𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎦−8𝑟⎡⎢⎢⎢⎢⎣𝐴³𝐵^∗1𝐴₁𝑡⎤⎥⎥⎥⎥⎦+2𝑛,_2𝑖=𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝑁^1𝑖𝑀^𝑇1𝑖𝑀𝑁_31𝑇11𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁𝑀₁₁𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎦−4𝑟⎡⎢⎢⎢⎢⎣𝐴³𝐵^∗1𝐴₁⎤⎥⎥⎥⎥⎦−4𝑟⎡⎢⎢⎣𝐵_∗1𝐴₁⎤⎥⎥⎦+2𝑛.(2.35)

Proof. We only prove the case that 𝑖=1. Similarly, we can get the results that 𝑖=2,3,4. Let 14𝑃¹𝜙𝑋₀𝑃^𝑇1+14𝑃²𝜙𝑋₀𝑃^𝑇2+14𝑃³𝜙𝑋₀𝑃^𝑇3+14𝑃⁴𝑃𝜙𝑋₀𝑃^𝑇4=𝐴,₁,𝑃₂,⎡⎢⎢⎢⎢⎢⎢⎣𝑃𝐿₃,𝑃₄𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)=𝐵,_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4⎤⎥⎥⎥⎥⎥⎥⎦=𝐶;(2.36) note that 𝐿_𝑀 is Hermtian; then 𝐿_𝜙(𝑀) is symmetric; hence (2.15) can be written as 𝑋₁=𝐴+𝐵𝑉𝐶+(𝐵𝑉𝐶)^∗.(2.37) Note that 𝐴=𝐴^∗ and ℛ(𝐵)⊆ℛ(𝐶^∗); applying (2.9) and (2.10) to (2.37) yields max_{𝑋1∈𝑆1}𝑟𝑋₁=min⎧⎪⎨⎪⎩𝑟𝐴,𝐶^∗,𝑟⎡⎢⎢⎣𝐵𝐴𝐵_∗0⎤⎥⎥⎦⎫⎪⎬⎪⎭,(2.38)min_{𝑋1∈𝑆1}𝑟𝑋₁=2𝑟𝐴,𝐶^∗+𝑟⎡⎢⎢⎣𝐵𝐴𝐵_∗0⎤⎥⎥⎦−2𝑟⎡⎢⎢⎣𝐶𝐴𝐵0⎤⎥⎥⎦.(2.39) Let 𝑎𝑃₁,𝑃₂,𝑃₃,𝑃₄=𝑃,_𝑖=⎡⎢⎢⎢⎢⎢⎢⎣𝜙𝐴^𝑖0𝜙0𝐴_𝑖0000𝜙0𝐴_𝑖0000𝜙0𝐴_𝑖𝑏⎤⎥⎥⎥⎥⎥⎥⎦,𝑖=1,3,₁=⎡⎢⎢⎢⎢⎢⎢⎣𝜙𝐵¹0𝜙0𝐵₁0000𝜙0𝐵₁0000𝜙0𝐵₁⎤⎥⎥⎥⎥⎥⎥⎦.(2.40) Note that 𝜙(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 𝜙(⋅) that 𝑟𝐴,𝐶^∗=𝑟⎡⎢⎢⎢⎢⎣𝐴0𝑏𝑃^𝑇10𝑎₁⎤⎥⎥⎥⎥⎦−4𝑟𝜙𝐴₁=𝑟⎡⎢⎢⎢⎢⎢⎣−10𝑃−4𝑟[𝜙(𝐹)]4𝜙^𝑇𝐶₂𝑃^𝑇1𝑏^𝑇1−14𝜙𝐶¹𝑃^𝑇1𝑎₁⎤⎥⎥⎥⎥⎥⎦−4𝑟⎡⎢⎢⎣𝜙𝐵_∗1𝜙𝐴₁=𝑟⎡⎢⎢⎢⎢⎣0𝑃₁⎤⎥⎥⎦𝜙,0,0,0_𝑇𝐶₂𝑃^𝑇1𝑏^𝑇1𝜙𝐶₁𝑃^𝑇1𝑎₁⎤⎥⎥⎥⎥⎦−4𝑟⎡⎢⎢⎣𝜙𝐵_∗1𝜙𝐴₁=𝑟⎡⎢⎢⎣𝐿₂₁⎤⎥⎥⎦𝑁^𝑇1𝑖𝐿₁₁𝑀_1𝑖⎤⎥⎥⎦−4𝑟⎡⎢⎢⎣𝜙𝜙𝐵𝐴^∗1₁⎤⎥⎥⎦+3𝑟⎡⎢⎢⎣𝜙𝐵^∗1𝜙𝐴₁=𝑟⎡⎢⎢⎣𝐿₂₁⎤⎥⎥⎦+𝑛𝑁^𝑇1𝑖𝐿₁₁𝑀_1𝑖⎤⎥⎥⎦−4𝑟⎡⎢⎢⎣𝐵^∗1𝐴₁⎤⎥⎥⎦+𝑛.(2.41) Note that 𝐿_𝐴=𝑅_𝐴^∗, then 𝐿_𝜙(𝐴)=𝑅_𝜙^∗_(𝐴); hence𝑟⎡⎢⎢⎣𝐵𝐴𝐵_∗0⎤⎥⎥⎦=𝑟⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑃𝐴𝑃000^𝑇0𝑎^𝑇3𝑏₁𝑎^𝑇100𝑏𝑎_3𝑇10000𝑎₁000000⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦−8𝑟[𝜙(𝑀)]−8𝑟[𝜙(𝐹)]−8𝑟𝜙𝐴¹=𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝑀_𝑇31𝑁₁₁𝑀^𝑇11𝑀𝑁_31𝑇11𝜙𝐶₃𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁𝑀₁₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₃𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁−8𝑟⎡⎢⎢⎢⎢⎣𝜙𝐴𝜙₃𝐶₁𝜙^𝑇𝐴₃𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎦𝜙𝐵_∗1𝜙𝐴₁⎤⎥⎥⎥⎥⎦+6𝑟⎡⎢⎢⎢⎢⎣𝜙𝐴³𝜙𝐵_∗1𝜙𝐴₁=𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝑀_𝑇31⎤⎥⎥⎥⎥⎦+2𝑛𝑁₁₁𝑀^𝑇11𝑀𝑁_31𝑇11𝜙𝐶₃𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁𝑀₁₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₃𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁𝜙𝐶₁𝜙^𝑇𝐴₃𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎦−8𝑟⎡⎢⎢⎢⎢⎣𝐴³𝐵^∗1𝐴₁⎤⎥⎥⎥⎥⎦+2𝑛.(2.42) Similarly, we can obtain 𝑟⎡⎢⎢⎣𝐶𝐴𝐵0⎤⎥⎥⎦=𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝑁¹¹𝑀^𝑇11𝑀𝑁_31𝑇11𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁𝑀₁₁𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎦−4𝑟⎡⎢⎢⎢⎢⎣𝐴³𝐵^∗1𝐴₁⎤⎥⎥⎥⎥⎦−4𝑟⎡⎢⎢⎣𝐵_∗1𝐴₁⎤⎥⎥⎦+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 2𝑟⎡⎢⎢⎣𝐿²¹𝑁^𝑇1𝑖𝐿₁₁𝑀_1𝑖⎤⎥⎥⎦+𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝑀^𝑇31𝑁₁₁𝑀^𝑇11𝑀𝑁_31𝑇11𝜙𝐶₃𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁𝑀₁₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₃𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁=2𝑟⎡⎢⎢⎢⎢⎢⎢⎣0𝑁𝜙_1𝑖𝐶₁𝜙^𝑇𝐴₃𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎦𝑀^𝑇1𝑖𝑀𝑁_31𝑇11𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁𝑀₁₁𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎦(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+𝑋₄𝑘, 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 𝑋_𝑖=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,𝐵₁,𝐶₁,𝐶₂, and 𝐶₃ be defined in Lemma 2.2, 𝐴₄∈ℍ^𝑙×𝑛,𝐶₄∈ℍ^𝑙×𝑙, and suppose that system (1.5) and the matrix equation 𝐴₄𝑌𝐴^∗4=𝐶₄ over ℍ have Hermitian solutions 𝑋 and 𝑌∈ℍ^𝑛×𝑛, 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 𝑟⎡⎢⎢⎣0𝑀^𝑇31𝑀₃₁𝜙𝐶₃⎤⎥⎥⎦=2𝑟𝑀³¹𝑟⎡⎢⎢⎢⎢⎣0𝑀_𝑇41,(3.1)0𝑀_𝑇41100𝑀₄₁𝜙𝐵₁𝜙^𝑇𝐴₁𝜙𝐶₄𝜙𝐴₄𝜙𝐶₂𝜙𝐴₄𝜙^𝑇𝐶₁⎤⎥⎥⎥⎥⎦=𝑟𝑀⁴¹+𝑟⎡⎢⎢⎣𝑀^𝑇41𝑀00_𝑇411𝜙𝐵₁𝜙^𝑇𝐴₁⎤⎥⎥⎦,(3.2)𝑟⎡⎢⎢⎢⎢⎢⎢⎣00𝑀^𝑇41𝑀0₄₁𝜙𝑀_411𝑇𝜙𝐶₄0𝜙𝐴𝐵₁₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₄𝜙𝐶₁𝜙^𝑇𝐴₄⎤⎥⎥⎥⎥⎥⎥⎦=𝑟𝑀⁴¹+𝑟⎡⎢⎢⎢⎢⎣𝑀⁴¹𝑀₄₁₁00𝜙𝜙^𝑇𝐴𝐵₁₁𝑟⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣00𝑀_𝑇41⎤⎥⎥⎥⎥⎦,00𝑀_𝑇411000𝑀₄₁𝜙^𝑇𝐴₃𝜙𝐵₁𝜙^𝑇𝐴₁0𝜙𝑀𝐴₄₁₁₃𝜙𝐶₄0000𝜙_𝑇0𝜙𝐶₃𝜙𝐴₃𝜙𝐶₂𝜙𝐴₃𝜙^𝑇𝐶₁0𝜙𝐴𝐵₁₁0𝜙^𝑇𝐶₂𝜙^𝑇𝐴₃𝜙^𝑇𝐶₂𝜙𝐵₁𝜙^𝑇𝐶₂𝜙^𝑇𝐴₁0𝜙𝐶₁𝜙^𝑇𝐴₃𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=2𝑟⎡⎢⎢⎢⎢⎢⎢⎣𝑀⁴¹𝑀₄₁₁00𝜙𝜙_𝑇𝐴₃0𝜙𝐴𝐵₁₁𝑟⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣00𝑀_𝑇41⎤⎥⎥⎥⎥⎥⎥⎦,00𝑀_𝑇41100𝑀₄₁𝜙𝐵₁𝜙^𝑇𝐴₁0𝜙𝑀_411𝑇𝜙𝐶₄000𝜙𝐴𝐵₁₁0𝜙^𝑇𝐵₁𝜙𝐶₂𝜙^𝑇𝐵₁𝜙^𝑇𝐶₁0𝜙𝐶₁𝜙𝐵₁𝜙𝐶₁𝜙^𝑇𝐴₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=2𝑟⎡⎢⎢⎢⎢⎣𝑀⁴¹𝑀₄₁₁00𝜙𝜙^𝑇𝐴𝐵₁₁⎤⎥⎥⎥⎥⎦,(3.3) where 𝑀₄₁=⎡⎢⎢⎢⎢⎢⎢⎣𝐴⁴²𝐴₄₃𝐴₄₄𝐴₄₁𝐴₄₄−𝐴₄₃−𝐴₄₄𝐴₄₁𝐴₄₂𝐴₄₃−𝐴₄₂𝐴₄₁⎤⎥⎥⎥⎥⎥⎥⎦,𝑀₄₁₁=⎡⎢⎢⎢⎢⎢⎢⎣𝐴²¹−𝐴₂₂000−𝐴₂₃000−𝐴₂₄000000⎤⎥⎥⎥⎥⎥⎥⎦.(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=14𝑃¹𝜙𝑋₀𝑃^𝑇1+14𝑃²𝜙𝑋₀𝑃^𝑇2+14𝑃³𝜙𝑋₀𝑃^𝑇3+14𝑃⁴𝜙𝑋₀𝑃^𝑇4+𝑃₁,𝑃₂,𝑃₃,𝑃₄𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉⎡⎢⎢⎢⎢⎢⎢⎣𝐿^𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4+𝑃₁⎤⎥⎥⎥⎥⎥⎥⎦,𝑃₂,𝑃₃,𝑃₄𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝑉^𝑇⎡⎢⎢⎢⎢⎢⎢⎣𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝑀)𝐿_𝜙(𝐹)𝐿_𝜙(𝐴1)𝑃^𝑇4⎤⎥⎥⎥⎥⎥⎥⎦,(3.5) where 𝑉 is arbitrary matrices with compatible sizes.
Let 𝐴₁,𝐶₁=0;𝐵₁,𝐶₂=0;𝐴₃=𝐴₄; 𝐶₃=𝐶₄ in Corollary 2.8 and (2.15). It is easy to verify that the matrix equation 𝐴₄𝑌𝐴^∗4=𝐶₄ over ℍ has a real Hermitian solution if and only if (3.1) hold and the real Hermitian solution can be expressed as 𝑌₁=14𝑃¹𝜙𝑌₀𝑃^𝑇1+14𝑃²𝜙𝑌₀𝑃^𝑇2+14𝑃³𝜙𝑌₀𝑃^𝑇3+14𝑃⁴𝜙𝑌₀𝑃^𝑇4+𝑃₁,𝑃₂,𝑃₃,𝑃₄𝐿_𝜙(𝐴4)𝑈+𝑈^𝑇⎡⎢⎢⎢⎢⎢⎢⎣𝐿_𝜙(𝐴1)𝑃^𝑇1𝐿_𝜙(𝐴1)𝑃^𝑇2𝐿_𝜙(𝐴1)𝑃^𝑇3𝐿_𝜙(𝐴1)𝑃^𝑇4⎤⎥⎥⎥⎥⎥⎥⎦,(3.6) where 𝜙(𝑌₀) is a particular solution to 𝜙(𝐴₄)(𝑌_𝑖𝑗)_4×4𝜙^𝑇(𝐴₄)=𝜙(𝐶₄) and 𝑈 is arbitrary matrices with compatible sizes. The expression of 𝑌₁ can also be obtained from Lemma 2.1. Let 𝐺=1𝑃₁,𝑃₂,𝑃₃,𝑃₄=𝑃,4𝑃¹𝜙𝑋₀𝑃^𝑇1+14𝑃²𝜙𝑋₀𝑃^𝑇2+14𝑃³𝜙𝑋₀𝑃^𝑇3+14𝑃⁴𝜙𝑋₀𝑃^𝑇4−14𝑃¹𝜙𝑌₀𝑃^𝑇1−14𝑃²𝜙𝑌₀𝑃^𝑇2−14𝑃³𝜙𝑌₀𝑃^𝑇3−14𝑃⁴𝜙𝑌₀𝑃^𝑇4.(3.7) Equating 𝑋₁ and 𝑌₁, we obtain the following equation: 𝑋₁−𝑌₁=𝐺+𝑃𝐿_𝜙(𝐴1)𝐿_𝜙(𝐹)𝐿_𝜙(𝑀)𝑉⎡⎢⎢⎢⎢⎢⎢⎣𝐿^𝜙(𝐹)𝐿