﻿ 弱双四元数矩阵方程AXA<sup>H</sup>+BYB<sup>H</sup>=C的反Hermite解

# 弱双四元数矩阵方程AXAH+BYBH=C的反Hermite解On Anti-Hermitian Solutions of the Reduced Biquaternion Matrix Equation AXAH+BYBH=C

Abstract: In this paper, we discuss Anti-Hermitian solutions of reduced biquaternion matrix equation AXAH+BYBH=C , where A,B are known reduced biquaternion matrices with suitable size, C is a known reduced biquaternion anti-Hermitian matrix with suitable size, and X,Y are unknown re-duced biquaternion anti-Hermitian square matrices with suitable size. The objective of this paper is to establish a necessary and sufficient condition for the existence of a solution and a solution expression.

1. 引言

$AX{A}^{H}+BY{B}^{H}=C$ (1)

${H}_{E}=\left\{X,Y|X,Y\in AH{Q}_{RB}^{n×n},AX{A}^{H}+BY{B}^{H}=C\right\}$ (2)

$q={q}_{0}+{q}_{1}i+{q}_{2}j+{q}_{3}k$ (3)

$\stackrel{¯}{q}={q}_{0}-{q}_{1}i-{q}_{2}j-{q}_{3}k$ (4)

q还可以表示为

$q={c}_{1}+{c}_{2}j$ ,

q的复数矩阵表示为

$f\left(q\right)=\left[\begin{array}{cc}{c}_{1}& {c}_{2}\\ {c}_{2}& {c}_{1}\end{array}\right]$ .

$q,{q}^{\prime }\in {Q}_{RB}$ ，显然有 $f\left(q{q}^{\prime }\right)=f\left(q\right)f\left({q}^{\prime }\right)$

$F\left(A\right)=\left[\begin{array}{cc}{A}_{1}& {A}_{2}\\ {A}_{2}& {A}_{1}\end{array}\right]\in {C}^{2m×2n}$ .

$F\left(A\right)$ 是由A唯一决定。对于 $A\in {Q}_{RB}^{m×n},B\in {Q}_{RB}^{n×s}$ ，有 $F\left(AB\right)=F\left(A\right)F\left(B\right)$ 。由定义易知，矩阵A可以唯一表示为

$A=\mathrm{Re}\left({A}_{1}\right)+\mathrm{Im}\left({A}_{1}\right)i+\mathrm{Re}\left({A}_{2}\right)j+\mathrm{Im}\left({A}_{2}\right)k$ (5)

$\stackrel{¯}{A}=\mathrm{Re}\left({A}_{1}\right)-\mathrm{Im}\left({A}_{1}\right)i-\mathrm{Re}\left({A}_{2}\right)j-\mathrm{Im}\left({A}_{2}\right)k$ .

$\left(A,B,C\right)\otimes D=\left(A\otimes D,B\otimes D,C\otimes D\right)$$\left(\begin{array}{cc}E& F\\ G& H\end{array}\right)\otimes K=\left(\begin{array}{cc}E\otimes K& F\otimes K\\ G\otimes K& H\otimes K\end{array}\right)$ .

$vec\left(A\right)={\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)}^{\text{T}}$ .

${\Phi }_{A}=\left[{A}_{1},{A}_{2}\right]$ .

i) $A=B$ 当且仅当 ${\Phi }_{A}={\Phi }_{B}$ ；ii) ${\Phi }_{A+B}={\Phi }_{A}+{\Phi }_{B}$${\Phi }_{kA}=k{\Phi }_{A}$ ；iii) ${\Phi }_{AC}={\Phi }_{A}F\left(C\right)$

2. $vec\left({\Phi }_{AXB}\right)$ 的结构

$vec\left(ABC\right)=\left({C}^{\text{T}}\otimes A\right)vec\left(B\right)$ (6)

$vec\left({\Phi }_{ABC}\right)=\left[\begin{array}{cc}{C}_{1}^{\text{T}}\otimes {A}_{1}+{C}_{2}^{\text{T}}\otimes {A}_{2}& {C}_{2}^{\text{T}}\otimes {A}_{1}+{C}_{1}^{\text{T}}\otimes {A}_{2}\\ {C}_{2}^{\text{T}}\otimes {A}_{1}+{C}_{1}^{\text{T}}\otimes {A}_{2}& {C}_{1}^{\text{T}}\otimes {A}_{1}+{C}_{2}^{\text{T}}\otimes {A}_{2}\end{array}\right]\left[\begin{array}{c}vec\left({B}_{1}\right)\\ vec\left({B}_{2}\right)\end{array}\right]$ (7)

$ve{c}_{S}\left(A\right)={\left({a}_{1},{a}_{2},\cdots ,{a}_{n-1},{a}_{n}\right)}^{\text{T}}\in {Q}_{RB}^{\frac{n\left(n+1\right)}{2}}$ (8)

$ve{c}_{A}\left(B\right)=\sqrt{2}{\left({b}_{1},{b}_{2},\cdots ,{b}_{n-2},{b}_{n-1}\right)}^{\text{T}}\in {Q}_{RB}^{\frac{n\left(n-1\right)}{2}}$ (9)

i)

$X\in S{R}^{n×n}⇔vec\left(X\right)={K}_{S}ve{c}_{S}\left(X\right)$ (10)

${K}_{S}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccccccccccccc}\sqrt{2}{e}_{1}& {e}_{2}& \cdots & {e}_{n-1}& {e}_{n}& 0& 0& \cdots & 0& \cdots & 0& 0& 0\\ 0& {e}_{1}& \cdots & 0& 0& \sqrt{2}{e}_{2}& {e}_{3}& \cdots & {e}_{n}& \cdots & 0& 0& 0\\ 0& 0& \cdots & 0& 0& 0& {e}_{2}& \cdots & 0& \cdots & 0& 0& 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮& ⋮& & ⋮& & ⋮& ⋮& ⋮\\ 0& 0& \cdots & {e}_{1}& 0& 0& 0& \cdots & 0& \cdots & \sqrt{2}{e}_{n-1}& {e}_{n}& 0\\ 0& 0& \cdots & 0& {e}_{1}& 0& 0& \cdots & {e}_{2}& \cdots & 0& {e}_{n-1}& \sqrt{2}{e}_{n}\end{array}\right]$ ,

ii)

$X\in AS{R}^{n×n}⇔vec\left(X\right)={K}_{A}ve{c}_{A}\left(X\right)$ (11)

${K}_{A}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccccccccccc}{e}_{2}& {e}_{3}& \cdots & {e}_{n-1}& {e}_{n}& 0& \cdots & 0& 0& \cdots & 0\\ -{e}_{1}& 0& \cdots & 0& 0& {e}_{3}& \cdots & {e}_{n-1}& {e}_{n}& \cdots & 0\\ 0& -{e}_{1}& \cdots & 0& 0& -{e}_{2}& \cdots & 0& 0& \cdots & 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮& & 0& 0& & 0\\ 0& 0& \cdots & -{e}_{1}& 0& 0& \cdots & -{e}_{2}& 0& \cdots & {e}_{n}\\ 0& 0& \cdots & 0& -{e}_{1}& 0& \cdots & 0& -{e}_{2}& \cdots & -{e}_{n-1}\end{array}\right]$ ,

$M=\left[\begin{array}{cccc}{K}_{A}& i{K}_{S}& 0& 0\\ 0& 0& {K}_{S}& i{K}_{S}\end{array}\right]$ .

$\left[\begin{array}{c}vec\left({X}_{1}\right)\\ vec\left({X}_{2}\right)\end{array}\right]=M\left[\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({X}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\right]$ . (12)

$\begin{array}{c}\left[\begin{array}{c}vec\left({X}_{1}\right)\\ vec\left({X}_{2}\right)\end{array}\right]=\left[\begin{array}{c}vec\left(\mathrm{Re}\left({X}_{1}\right)\right)+ivec\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ vec\left(\mathrm{Re}\left({X}_{2}\right)\right)+ivec\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\right]=\left[\begin{array}{c}{K}_{A}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)+i{K}_{S}ve{c}_{S}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ {K}_{S}ve{c}_{S}\left(\mathrm{Re}\left({X}_{2}\right)\right)+i{K}_{S}ve{c}_{S}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\right]\\ =M\left[\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({X}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\right]\end{array}$ .

$vec\left({\Phi }_{AXB}\right)=\left[\begin{array}{cc}{B}_{1}^{\text{T}}\otimes {A}_{1}+{B}_{2}^{\text{T}}\otimes {A}_{2}& {B}_{2}^{\text{T}}\otimes {A}_{1}+{B}_{1}^{\text{T}}\otimes {A}_{2}\\ {B}_{2}^{\text{T}}\otimes {A}_{1}+{B}_{1}^{\text{T}}\otimes {A}_{2}& {B}_{1}^{\text{T}}\otimes {A}_{1}+{B}_{2}^{\text{T}}\otimes {A}_{2}\end{array}\right]M\left[\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({X}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\right]$ (13)

3. 矩阵方程(1)的解

$A{A}^{+}b=b$ (14)

$x={A}^{+}b+\left({I}_{n}-{A}^{+}A\right)y$ (15)

${P}_{1}=\left[\begin{array}{cc}\stackrel{¯}{{A}_{1}}\otimes {A}_{1}-{A}_{2}\otimes {A}_{2}& -{A}_{2}\otimes {A}_{1}+\stackrel{¯}{{A}_{1}}\otimes {A}_{2}\\ -{A}_{2}\otimes {A}_{1}+\stackrel{¯}{{A}_{1}}\otimes {A}_{2}& \stackrel{¯}{{A}_{1}}\otimes {A}_{1}-{A}_{2}\otimes {A}_{2}\end{array}\right]M$ ,

${P}_{2}=\left[\begin{array}{cc}\stackrel{¯}{{B}_{1}}\otimes {B}_{1}-{B}_{2}\otimes {B}_{2}& -{B}_{2}\otimes {B}_{1}+\stackrel{¯}{{B}_{1}}\otimes {B}_{2}\\ -{B}_{2}\otimes {B}_{1}+\stackrel{¯}{{B}_{1}}\otimes {B}_{2}& \stackrel{¯}{{B}_{1}}\otimes {B}_{1}-{B}_{2}\otimes {B}_{2}\end{array}\right]M$ , $e=\left[\begin{array}{c}vec\left(\mathrm{Re}\left({C}_{1}\right)\right)\\ vec\left(\mathrm{Re}\left({C}_{2}\right)\right)\\ vec\left(\mathrm{Im}\left({C}_{1}\right)\right)\\ vec\left(\mathrm{Im}\left({C}_{2}\right)\right)\end{array}\right]$ .

$\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]{\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}e=e$ .

$\begin{array}{l}A{H}_{E}=\left\{X,Y|\begin{array}{c}\underset{}{\overset{}{}}\\ \end{array}\\ \stackrel{\to }{X}=\left(\begin{array}{cc}{M}_{n}& 0\end{array}\right){\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}e+\left(\begin{array}{cc}{M}_{n}& 0\end{array}\right)\left({I}_{2{n}^{2}+n+2{p}^{2}+p}+{\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]\right)y,\\ \stackrel{\to }{Y}=\left(\begin{array}{cc}0& {M}_{p}\end{array}\right){\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}e+\left(\begin{array}{cc}0& {M}_{p}\end{array}\right)\left({I}_{2{n}^{2}+n+2{p}^{2}+p}+{\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]\right)y\right\}\end{array}$

y是有合适维数的任意向量。进一步，当

$rank\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]=2{n}^{2}+n+2{p}^{2}+p$

$A{H}_{E}=\left\{X,Y|\stackrel{\to }{X}=\left(\begin{array}{cc}{M}_{n}& 0\end{array}\right){\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}e,\stackrel{\to }{Y}=\left(\begin{array}{cc}0& {M}_{p}\end{array}\right){\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}e\right\}$ .

$AX{A}^{H}+BY{B}^{H}=C⇔{\Phi }_{AX{A}^{H}}+{\Phi }_{BY{B}^{H}}={\Phi }_{C}⇔{P}_{1}\left[\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({X}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\right]+{P}_{2}\left[\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({Y}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{2}\right)\right)\end{array}\right]=\left[\begin{array}{c}vec\left({C}_{1}\right)\\ vec\left({C}_{2}\right)\end{array}\right]$

$⇔\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]\left[\begin{array}{c}\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({X}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\\ \begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({Y}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{2}\right)\right)\end{array}\end{array}\right]=e$ .

$\left[\begin{array}{c}\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({X}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\\ \begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({Y}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{2}\right)\right)\end{array}\end{array}\right]={\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}e+\left[{I}_{2{n}^{2}+n}-{\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]\right]y$ .

$rank\left(\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]\right)=2{n}^{2}+n$ ,

$\left[\begin{array}{c}\begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({X}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({X}_{2}\right)\right)\end{array}\\ \begin{array}{c}ve{c}_{A}\left(\mathrm{Re}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{1}\right)\right)\\ ve{c}_{s}\left(\mathrm{Re}\left({Y}_{2}\right)\right)\\ ve{c}_{s}\left(\mathrm{Im}\left({Y}_{2}\right)\right)\end{array}\end{array}\right]={\left[\begin{array}{cc}\mathrm{Re}\left({P}_{1}\right)& \mathrm{Re}\left({P}_{2}\right)\\ \mathrm{Im}\left({P}_{1}\right)& \mathrm{Im}\left({P}_{2}\right)\end{array}\right]}^{+}e$

4. 结论

[1] Hammarling, S.J. (1982) Numerical Solution of the Stable, Non-Negative Definite Lyapunov Equation. IMA Journal of Numerical Analysis, 2, 303-323.
https://doi.org/10.1093/imanum/2.3.303

[2] Khatri, C.G. and Mitra, S.K. (1976) Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations. SIAM Journal on Applied Mathematics, 31, 579-585.
https://doi.org/10.1137/0131050

[3] Schütte, H.D. and Wenzel, J. (1990) Hypercomplex Numbers in Digital Signal Processing. IEEE International Symposium on Circuits and Systems, 2, 1557-1560.
https://doi.org/10.1109/ISCAS.1990.112431

[4] Hamilton, W.R. (1866) Elements of Quaternions. Longmans, London.

[5] Pei, S.C., Chang, J.H., Ding, J.J. and Chen, M.Y. (2008) Eigenvalues and Singular Value Decomposi-tions of Reduced Biquaternion Matrices. IEEE Transactions on Circuits and Systems I, 55, 2673-2685.
https://doi.org/10.1109/TCSI.2008.920068

[6] Yuan, S.F., Liao, A.P. and Lei, Y. (2008) Least Squares Her-mitian Solution of the Matrix Equation (AXB, CXD) = (E, F) with the Least Norm over the Skew Field of Quaternions. Mathematical and Computer Modelling, 48, 91-100.
https://doi.org/10.1016/j.mcm.2007.08.009

[7] Ben-Israle, A. and Greville, T.N.E. (2003) Generalized Inverses: Theory and Applications. Springer, New York.

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