Consimilarity and quaternion matrix equations $AX-\hat{X}B=C$, $X-A\hat{X}B=C$

L.Huang [Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix $A$ with respect to consimilarity transformations $\tilde{S}^{-1}AS$ in which $S$ is a nonsingular quaternion matrix and $\tilde{h}:=a-bi+cj-dk$ for each quaternion $h=a+bi+cj+dk$. We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations $\hat{S}^{-1}AS$ in which $h\mapsto\hat{h}$ is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$.


Introduction
We give a canonical form of a square quaternion matrix with respect to consimilarity transformations A ↦Ŝ −1 AS in which S is a nonsingular quaternion matrix and α ↦α is a fixed involutive automorphism of the skew field of quaternions H, and apply it to the quaternion matrix equations AX −XB = C and X − AXB = C.
A canonical form of a square complex matrix A with respect to consimilarity transformations A ↦S −1 AS (S ∈ C n×n is nonsingular) (1) was given by Hong and Horn [4] (see also [5,Theorem 4.6.12]): A is consimilar to a direct sum, uniquely determined up to permutation of direct summands, of matrices of the following two types: in which is the k × k Jordan block with eigenvalue λ. Huang [7] defined consimilarity transformations of square quaternion matrices A ↦S −1 AS (S ∈ H n×n is nonsingular) (4) via the involution h = a + bi + cj + dk ↦h ∶= −jhj = a − bi + cj − dk (a, b, c, d ∈ R); on the skew field of quaternions H; we suggest to call (4) j-consimilarity transformations sinceS = −jSj. (The main reason why consimilarity transformations of quaternion matrices are not defined via the quaternion conju- The definition of consimilarity transformations via (5) looks special. We show in Lemma 1 that all consimilarity transformations defined via involutive automorphisms of H are transformed from one to each other by reselections of the set of orthogonal imaginary units i, j, k. Thus, we can use an involutive automorphism for which the formulas of the theory of consimilarity transformations are simpler.
Huang [7] and other authors (see, for example, [10,15,16,17,20]) study the j-consimilarity transformations (4). These transformations act on complex matrices as the consimilarity transformations (1). We suggest to define consimilarity transformations of quaternion matrices extending the similarity transformations of complex matrices. For this purpose, we define i-consimilarity transformationsŜ −1 AS of quaternion matrices via the automorphism Our goal is to show that i-consimilarity transformations are more convenient for use than j-consimilarity transformations (4): • Most properties of j-consimilarity transformations of quaternion matrices are closer to properties of similarity transformations of complex matrices than to properties of consimilarity transformations of complex matrices (see (6) and compare the canonical forms given in (2) and (7)). [20]), which admits us to study similarity and iconsimilarity transformations of quaternion matrices simultaneously in Sections 2 and 3.
• The canonical form in Theorem 3 of a quaternion matrix for iconsimilarity transformationsŜ −1 AS is a complex matrix (compare with (7)); in Section 3 we apply it to the quaternion matrix equations AX −XB = C and X − AXB = C, reducing them to complex matrix equations.

A canonical form for i-consimilarity
Consimilarity transformations come from the theory of semilinear operators (which is presented, for example, in Jacobson's book [8, Chapter 3, Section 12]). It is worth mention that the fundamental theorem of projective geometry is formulated in terms of semilinear maps between vector spaces. Let V be a right vector space over a field or skew field F with a fixed automorphism α ↦α on F.
. By transfer to other bases the matrix A is reduced by transformations For the most important types of automorphisms α ↦α, Jacobson [8, Chapter 3, Theorem 34] reduced the problem of classifying matrices with respect to transformations (8) to the problem of classifying matrices with respect to similarity. All automorphisms h ↦ĥ on H that we consider are involutive; that is,

Lemma 1.
If h ↦ĥ is an involutive automorphism of H, then either it is identical, or the set of orthogonal imaginary units i, j, k can be chosen such This automorphism is involutive if and only if σ −2 qσ 2 = q for all q ∈ H, if and only if σ 2 ∈ R. Multiplying σ by a positive number (which does not change the automorphism), we get σ 2 = ±1. Write σ = a + bτ , in which a, b ∈ R, τ ∉ R, and τ = 1. Since If σ 2 = 1, then b = 0, a 2 = 1, σ = a = ±1, and the automorphism q ↦ q σ is the identity.
Let σ 2 = −1. Then a = 0, b 2 = 1, b = ±1, and so σ = ±τ . Recall that the space of pure quaternions can be identified with the vector space R 3 ; the product of pure quaternions u and v can be represented in the form is the usual inner (scalar) product and u × v is the vector cross product. Reselecting the imaginary units i, j, k, we set i ∶= τ , take as j any pure quaternion such that j = 1 and (i, j) = 0 (i.e, j is any imaginary unit that is perpendicular to i), put k ∶= ij, and obtain the automorphism (9).
The automorphism (9) can be written in the form Thus, h σ = u + σ 2 v for σ ∈ {1, i} and all h ∈ H. This admits us to study similarity and consimilarity transformations simultaneously using the following definition.
Let σ ∈ {1, i}. By σ-consimilarity transformations of quaternion matrices we mean the transformations in which S is nonsingular. It suffices to study σ-consimilarity transformations since by Lemma 1 each involutive automorphism of H has the form h ↦ h σ = σ −1 hσ (σ ∈ {1, i}) in suitable i, j, k. (iv) Ai and Bi are similar.
Theorem 3 (cf. [7,Theorem 3]). Each square quaternion matrix A is σconsimilar (σ ∈ {1, i}) to a complex matrix that is a direct sum, uniquely determined up to permutation of summands, of Jordan blocks Proof. Wiegmann [19,Theorem 1] proved (see also Zhang's survey [21, Theorem 6.4]) that each square quaternion matrix A is similar to a direct sum, uniquely determined up to permutation of summands, of Jordan blocks J k (a + bi), a, b ∈ R, b ⩾ 0. Using Lemma 2, we get the desired Jordan canonical form of quaternion matrices for i-consimilarity.

Quaternion matrix equations
In this section, we consider the quaternion matrix equations in which A ∈ H m×m , B ∈ H n×n , C ∈ H m×n and h ↦ĥ is an arbitrary involutive automorphism of H. These equations for the automorphism (5) are studied in [10,15,16,17,20] mainly by means of the replacement of p × q quaternion matrices with the corresponding 2p × 2q complex matrices or 4p × 4q real matrices. We use the consimilarity canonical form from Theorem 3 (in a similar way, Bevis, Hall, and Hartwig [1] studied the complex matrix equation AX − XB = C using the consimilarity canonical form of complex matrices). By Lemma 1, we can suppose thatĥ = h σ for some σ ∈ {1, i} and all h ∈ H, and get the equations For all nonsingular S ∈ H m×m and R ∈ H n×n , these equations are equivalent to Taking S and R such that S −σ AS and R −σ BR are the complex canonical forms of A and B determined by Theorem 3, we obtain the matrix equations that are considered in the following simple theorem and its corollaries.
Theorem 4. Let the quaternion matrix equations (10) be given by complex matrices A and B and a quaternion matrix C = C 1 + C 2 j (C 1 , C 2 ∈ C m×n ).
Then the sets of solutions of the equations (10) consist of all matrices X = X 1 + X 2 j in which X 1 and X 2 are complex matrices satisfying for the first equation in (10) and, respectively, for the second equation in (10).
Proof. Write X = X 1 + X 2 j and C = C 1 + C 2 j (X 1 , X 2 , C 1 , C 2 ∈ C m×n ). Then X i = X 1 − X 2 j, X σ = X 1 + σ 2 X 2 j, and the equations (10) take the form Since A and B are complex matrices and jB =Bj, the quaternion matrix equations (13) are partitioned into two pairs of complex matrix equations (11) and (12).

Corollary 5. Let us apply Theorem 4 to the quaternion matrix equation
be two complex Jordan matrices (for instance, canonical forms from Theorem 3 of quaternion matrices with respect to σ-consimilarity). Write (a) The following statements hold: • If M σ = ∅, then AX − X σ B = C has a unique solution.
• If M σ ≠ ∅, then two cases may arise: either AX − X σ B = C has no solutions, or the set of its solutions is infinite and consists of all matrices X ○ + Y in which X ○ is a fixed particular solution of AX − X σ B = C and Y runs over all solutions of AY − Y σ B = 0.
(b) The set of solutions of AY − Y σ B = 0 is described as follows. Let us partition Y into blocks in accordance with the partitions of A and B: Then the set of solutions of AY − Y σ B = 0 consists of all quaternion matrices of the form U + V j, in which q β=1 are complex matrices that are partitioned conformally to Y , • the other U αβ and V αβ have the form if the number of rows is less than or equal to the number of columns or, respectively, the number of rows is greater than the number of columns (we write the off-diagonal units in all Jordan blocks over the diagonal; see (3)).
Its set (15) is {i}. Its particular solution is To solve the corresponding equation in which the right-hand part is zero: we write Y = U + V j (U and V are complex matrices) and partition it as in (16): Replacing these blocks by the corresponding pairs of eigenvalues (in the notation of Corollary 5(b)), we get By Corollary 5, the set of solutions of (18) consists of all matrices in which the stars denote complex numbers, and the set of solutions of (17) consists of all matrices Conclusion: In all the papers that we know, the consimilarity of quaternion matrices is defined via the automorphism h ↦ −jhj. We show that the consimilarity defined via h ↦ −ihi is more convenient for studying. All consimilarities of quaternion matrices defined via involutive automorphisms are reduced each other by reselecting of orthogonal imaginary units i, j, k in H.