1 Introduction

In 1849, James Cockle found the split quaternion, which has the following form:

$$\begin{aligned} q=q_1+q_2{\mathbf{i}}+q_3{\mathbf{j}}+q_4{\mathbf{k}},{\mathbf{i}}^2=-1,{\mathbf{j}}^2={\mathbf{k}}^2=1,{\mathbf{i}}{\mathbf{j}}{\mathbf{k}}=1, \end{aligned}$$

where \(q_1,q_2,q_3,q_4\) are real and \({\mathbf{i}}{\mathbf{j}}=-{\mathbf{j}}{\mathbf{i}}={\mathbf{k}},{\mathbf{j}}{\mathbf{k}}=-{\mathbf{k}}{\mathbf{j}}=-{\mathbf{i}},{\mathbf{k}}{\mathbf{i}}=-{\mathbf{i}}{\mathbf{k}}={\mathbf{j}}\). The set of all split quaternions, denoted by \({{\mathbf {S}}}{{\mathbf {Q}}}\), is an associative, noncommutative four-dimensional Clifford algebra and has zero divisors, nilpotent elements and nontrivial idempotents [2]. \({{\mathbf {S}}}{{\mathbf {Q}}}\) is different from the quaternion skew field and has more complicated algebraic structure. For details, see [2, 3, 13, 14] and the references therein.

In complexified classical and non-Hermitian quantum mechanics, there are surprising relations between quaternionic and split quaternionic mechanics [4]. In the literature over the past decade, the complexified mechanical systems with real energies are studied extensively, which can alternatively be viewed as certain split quaternionic extensions of the underlying real mechanical systems [19]. This result leads to the possibility of employing algebraic techniques of quaternions and split quaternions to deal with some challenging open issues in complexified classical quantum mechanics.

For split quaternion matrices, by means of various representations, Jiang, et al. [10,11,12, 21] studied algebraic methods for least squares problem, diagonalization, eigenvalues and eigenvectors, Schrödinger equations, respectively. Hidayet Hüda Kösal, et al. [1] discussed the consimilarity of split quaternions and Split Quaternion Matrices. Melek Erdoǧdu et al. [7, 8] studied exponential and eigenvalues of split quaternion matrices, respectively.

As far as we know, there is no article discussing the elementary transformation of the split quaternion matrix. Although the elementary transformation of the complex matrix is simple, the elementary transformation of the split quaternion matrix is slightly different, and it plays an important role in the determinant and inverse matrix, etc. In this paper, we will study the elementary transformation of the split quaternion matrix and initially discuss its applications in determinant and inverse matrix.

Next, we review some basic knowledge for split quaternion matrices. Let \({\mathbf {R}},{\mathbf {C}}\), \({\mathbf {H}}\) and \({\mathbf {F}}^{m\times n}\) denote the real number field, the complex number field, the quaternion skew field, the set of \(m\times n\) matrices on \({\mathbf {F}}\), respectively. For

$$\begin{aligned} a=a_1+a_2{\mathbf{i}}+a_3{\mathbf{j}}+a_4{\mathbf{k}}=(a_1+a_2{\mathbf{i}})+(a_3+a_4{\mathbf{i}}){\mathbf{j}} \triangleq {\hat{a}}_1+{\hat{a}}_2 {\mathbf{j}}\in {{\mathbf {S}}}{{\mathbf {Q}}} \end{aligned}$$

with \(a_i\in {\mathbf {R}}, i=1,2,3,4\) and \({\hat{a}}_j\in {\mathbf {C}},j=1,2\), the conjugate of a is defined as \(\bar{a}=a_1-a_2{\mathbf{i}}-a_3{\mathbf{j}}-a_4{\mathbf{k}}=\bar{{\hat{a}}}_1-{\hat{a}}_2 {\mathbf{j}}\), and then we have

$$\begin{aligned} a{\bar{a}} ={\bar{a}} a= a_1^2 +a_2^2-a_3^2-a_4^2. \end{aligned}$$

The modulus |a| of a split quaternion a is defined as

$$\begin{aligned} |a|^2= |a{\bar{a}}| = ||{\hat{a}}_1|^2-|{\hat{a}}_2|^2|=|a_1^2 +a_2^2-a_3^2-a_4^2|. \end{aligned}$$

It is easy to verify that

$$\begin{aligned} {\overline{ab}}={\bar{b}}{\bar{a}}, |ab|=|a||b|\,\, \text{ and }\,\,{\mathbf{j}}c={\bar{c}} {\mathbf{j}} \end{aligned}$$

for \( b\in {{\mathbf {S}}}{{\mathbf {Q}}}\) and \( c\in {\mathbf {C}}\). a is said to be a unit split quaternion if its modulus is 1. The inverse of the split quaternion a is

$$\begin{aligned} a^{-1}=\frac{{\bar{a}}}{{\bar{a}} a},\,\text{ if }\, |a|\not = 0. \end{aligned}$$

For \(a={\hat{a}}_1+{\hat{a}}_2 {\mathbf{j}}\in {{\mathbf {S}}}{{\mathbf {Q}}}\) with \({\hat{a}}_1,{\hat{a}}_2\in {\mathbf {C}}\), we call \({\hat{a}}_1\) the complex part, \({\hat{a}}_2\) the imaginary part, and denote them as \({\mathbf{comp}}(a)\) and \({\mathbf{imag}}(a)\) respectively. For any matrix A, A(i, j) and A(i : j, k : l) represent the (i, j)-element of A and the submatrix of A containing the intersection of rows i to j and columns k to l, respectively. \(I_n\) denotes the unit matrix of order n and \(e_i\) is the i-th column of I.

The following representations are usually used.

Theorem 1.1

([20]) For any split quaternion \(a=a_1+a_2{\mathbf{i}}+a_3{\mathbf{j}}+a_4{\mathbf{k}}\), the R in

$$\begin{aligned} a\rightarrow a^{\mathcal {R}}\triangleq \left( \begin{array}{cc}a_1+a_3&{}-a_2+a_4\\ a_2+a_4&{}a_1-a_3\end{array}\right) \in {\mathbf {R}}^{2\times 2} \end{aligned}$$
(1.1)

is an isomorphic mapping form \({{\mathbf {S}}}{{\mathbf {Q}}}\) to \({\mathbf {R}}^{2\times 2}\).

For any \(A=(a_{ij})\in {{\mathbf {S}}}{{\mathbf {Q}}}^{m\times n}\), let \(A^{\mathcal {R}}=(A_{ij})\) be a \(m\times n\) block matrix with \(A_{ij}=a_{ij}^{\mathcal {R}}\). Then the \({\mathcal {R}}\) in \(A\rightarrow A^{\mathcal {R}}\) is a one-to-one mapping from \({{\mathbf {S}}}{{\mathbf {R}}}^{m\times n}\) to \({\mathbf {R}}^{2m\times 2n}\) satisfying

$$\begin{aligned} (A+B)^{\mathcal {R}}=A^{\mathcal {R}}+B^{\mathcal {R}},(AC)^{\mathcal {R}}=A^{\mathcal {R}} C^{\mathcal {R}},(\lambda A)^{\mathcal {R}}=\lambda A^{\mathcal {R}}, \end{aligned}$$
(1.2)

where \(B\in {{\mathbf {S}}}{{\mathbf {Q}}}^{m\times n},C\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times s}, \lambda \in {\mathbf {R}}.\)

In [2], for any split quaternion \(a={\hat{a}}_1+{\hat{a}}_2 {\mathbf{j}}\) with \({\hat{a}}_1,{\hat{a}}_2\in {\mathbf {C}}\), its complex representation is defined as

$$\begin{aligned} a^{\mathcal {C}}\triangleq \left( \begin{array}{cc}{\hat{a}}_1&{}{\hat{a}}_2\\ \bar{{\hat{a}}}_2&{}\bar{{\hat{a}}}_1\end{array}\right) \in {\mathbf {C}}^{2\times 2}. \end{aligned}$$
(1.3)

For any split quaternion matrix \(A=(a_{ij})={\hat{A}}_1+{\hat{A}}_2 {\mathbf{j}}\) with \({\hat{A}}_1,{\hat{A}}_2\in {\mathbf {C}}^{m\times n}\), similar to (1.3), we define its complex representation as

$$\begin{aligned} A^{\mathcal {C}}=(a_{ij}^{\mathcal {C}})\in {\mathbf {C}}^{2m\times 2n}, \end{aligned}$$

which can be obtained from the complex representation

$$\begin{aligned} \chi _A=\left( \begin{array}{cc}{\hat{A}}_1&{}{\hat{A}}_2\\ \bar{{\hat{A}}}_2&{}\bar{{\hat{A}}}_1\end{array}\right) \end{aligned}$$

of [2] by row and column interchanges. But the former is more convenient for the discussion of this paper. We can prove that (1.2) is also true for the above two complex representations, \({\det }(a^{\mathcal {R}})={\det }(a^{\mathcal {C}})\) and

$$\begin{aligned} |a|^2=|{\det }(a^{\mathcal {R}})|=|{\det }(a^{\mathcal {C}})|. \end{aligned}$$

This paper is organized as follows. In Sect. 2, some basic results on the elementary transformation are discussed and the step of triangulating a split quaternion matrix is given. In Sect. 3, the determinants and inverse matrix of the split quaternion matrix are studied. Some concluding remarks are given in Sect. 4.

2 Elementary Transformation and Triangulation

In this section, we discuss the elementary transformation for the split quaternion matrix, which is basically the same as that for the complex matrix.

The following three elementary row operations applied to the split quaternion matrix yield an upper triangular matrix.

  1. (a)

    Interchanges: Interchanging the i-th row and the j-th row.The corresponding elementary matrix is denoted as P(i, j).

  2. (b)

    Scaling: Multiplying the i-th row by the split quaternion \(\lambda \) with \(|\lambda |\not =0\) from the left.The corresponding elementary matrix is denoted as \(P(\lambda *i)\).

  3. (c)

    Replacement: Multiplying the j-th row by the split quaternion \(\lambda \) from the left, then adding to the i-th row. The corresponding elementary matrix is denoted as \(P(i,\lambda *j)\).

The following result is important for triangulating a split quaternion matrix.

Lemma 2.1

For nonzero split quaternions \(a={\hat{a}}_1+{\hat{a}}_2 {\mathbf{j}}\) and \(b={\hat{b}}_1+{\hat{b}}_2 {\mathbf{j}}\) with \({\hat{a}}_1,{\hat{a}}_2,{\hat{b}}_1,{\hat{b}}_2\in {\mathbf {C}}\) and \(|a|=|b|=0\), if \(\frac{{\hat{b}}_1}{{\hat{a}}_1}\not =\frac{{\hat{b}}_2}{{\hat{a}}_2}\), then either \(|a+b|\not =0\) or \(|a+{\mathbf{i}}b|\not =0\) holds.

Proof

First, it is easy to know that

$$\begin{aligned} 0=|a|^2=|{\hat{a}}_1|^2-|{\hat{a}}_2|^2,0=|b|^2=|{\hat{b}}_1|^2-|{\hat{b}}_2|^2, {\hat{a}}_i\not =0,{\hat{b}}_i\not =0,i=1,2. \end{aligned}$$

Let \(t_1=\frac{{\hat{b}}_1}{{\hat{a}}_1}, t_2=\frac{{\hat{b}}_2}{{\hat{a}}_2}.\) If real parts of \(t_1\) and \(t_2\) are different,

$$\begin{aligned} |a+b|^2= & {} ||{\hat{a}}_1+{\hat{b}}_1|^2-|{\hat{a}}_2+{\hat{b}}_2|^2|\\= & {} |({\hat{a}}_1+{\hat{b}}_1)(\bar{{\hat{a}}}_1+\bar{{\hat{b}}}_1) -({\hat{a}}_2+{\hat{b}}_2)(\bar{{\hat{a}}}_2+\bar{{\hat{b}}}_2)|\\= & {} |{\hat{a}}_1\bar{{\hat{b}}}_1+{\hat{b}}_1\bar{{\hat{a}}}_1 -{\hat{a}}_2\bar{{\hat{b}}}_2-{\hat{b}}_2\bar{{\hat{a}}}_2| =|{\bar{t}}_1|{\hat{a}}_1|^2+t_1|{\hat{a}}_1|^2-{\bar{t}}_2| {\hat{a}}_2|^2-t_2|{\hat{a}}_2|^2|\\= & {} |({\bar{t}}_1+ t_1)-({\bar{t}}_2+ t_2)||{\hat{a}}_2|^2\not =0. \end{aligned}$$

For similar reason, if imaginary parts of \(t_1\) and \(t_2\) are different,

$$\begin{aligned}&|a+{\mathbf{i}}b|^2=|{\mathbf{i}}[( t_1- {\bar{t}}_1)-( t_2-{\bar{t}}_2)]||{\hat{a}}_2|^2\not =0. \end{aligned}$$

\(\square \)

The step of triangulating a matrix by the elementary transformation is basically the same as that in the real field. We only take a vector as an example. Let \(0\not =a=(a_1,a_2,\ldots ,a_n)^T \in {{\mathbf {S}}}{{\mathbf {Q}}}^{n}.\)

  1. (I)

    If \(|a_1|\not =0,\) we take

    $$\begin{aligned} L=\left( \begin{array}{cccc}1&{}0&{}\cdots &{}0\\ -a_2a_1^{-1}&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}\cdots &{}\vdots \\ -a_na_1^{-1}&{}0&{}\cdots &{}0\end{array}\right) \end{aligned}$$

    and then \(La=(a_1,0,\dots ,0)^T.\)

  2. (II)

    If \(|a_1|=0\) and \(|a_i|\not =0\), then we perform (I) on P(1, i)a.

  3. (III)

    If \(a_1\not =0, |a_i|=0,i=1,2,\ldots ,n\) and

    $$\begin{aligned} \frac{{\mathbf{comp}}(a_i)}{{\mathbf{comp}}(a_1)}=\frac{{\mathbf{imag}}(a_i)}{{\mathbf{imag}}(a_1)}=t_i,i=1,2,\cdots ,n, \end{aligned}$$

    we take

    $$\begin{aligned} L=\left( \begin{array}{cccc}1&{}0&{}\cdots &{}0\\ -t_2&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}\cdots &{}\vdots \\ -t_n&{}0&{}\cdots &{}0\end{array}\right) \end{aligned}$$

    and then \(La=(a_1,0,\dots ,0)^T.\)

  4. (IV)

    If \(a_1\not =0,|a_i|=0,i=1,2,\cdots ,n\) and

    $$\begin{aligned} \frac{{\mathbf{comp}}(a_l)}{{\mathbf{comp}}(a_1)}\not =\frac{{\mathbf{imag}}(a_l)}{{\mathbf{imag}}(a_1)}\, \text{ for } \text{ some }\,\,l, \end{aligned}$$

    it follows from Lemma 2.1 that the modulus of the first element of \(P(1,1*l)a\) or \(P(1,{\mathbf{i}}*l)a\) is not zero. We perform (I) on \(P(1,1*l)a\) or \(P(1,{\mathbf{i}}*l)a\).

Through the above discussion, we can get the following conclusion.

Theorem 2.1

For \(A\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n}\), there exist permutation matrices or unit lower triangular matrices \(P_1,P_2,\ldots , P_s\) and an upper triangular matrix U such that

$$\begin{aligned} P_s\cdots P_2P_1A=U. \end{aligned}$$

As we know, for any \(A\in {\mathbf {F}}^{n\times n}\)(\({\mathbf {F}}\) may be \({\mathbf {R}}\),\({\mathbf {C}}\) or \({\mathbf {H}}\) ), there exit a permutation matrix P, an unit lower triangular matrix L and an upper triangular matrix U such that \(PA=LU.\) But for split quaternion matrices, this conclusion is not necessarily true. For example, let

$$\begin{aligned} P\left( \begin{array}{cc}1+{\mathbf{j}}&{}1\\ 1-{\mathbf{j}}&{}2\end{array}\right) =\left( \begin{array}{cc}1&{}0\\ l&{}1\end{array}\right) \left( \begin{array}{cc}u_1&{}u_2\\ 0&{}u_3\end{array}\right) . \end{aligned}$$

If \(P=I_2\), we have \(l(1+{\mathbf{j}})=1-{\mathbf{j}}\) and then \(l^{\mathcal {R}}(1+{\mathbf{j}})^{\mathcal {R}}=1-{\mathbf{j}}^{\mathcal {R}}\), that is,

$$\begin{aligned} \left( \begin{array}{cc}l_{11}&{}l_{12}\\ l_{21}&{}l_{22}\end{array}\right) \left( \begin{array}{cc}2&{}0\\ 0&{}0\end{array}\right) =\left( \begin{array}{cc}0&{}0\\ 0&{}2\end{array}\right) , \end{aligned}$$

which leads to contradiction! For the same reason, if \(P=\left( \begin{array}{cc}0&{}1\\ 1&{}0\end{array}\right) \), this also leads to contradiction!

3 Some Applications

3.1 Determinant

As an important tool, the determinant of real or complex matrices has been studied very deeply. For the determinant of the quaternion matrix, although it is very difficult, there are also many results and many people have given different definitions [5, 6, 9, 15, 16]. For the determinant of the split quaternion matrix, as far as we know, only in [2, 17, 18], the authors defined the q-determinant of A by

$$\begin{aligned} |A|_q={\det }(\chi _A). \end{aligned}$$

We can easily prove that \(|A|_q\in {\mathbf {R}}\) and \(||a|_q|=|a|^2\) for any split quaternion a.

Next, we give two new determinants.

Definition 1

For \(A\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n}\), we define two determinants as

$$\begin{aligned} {\mathrm{sdet}}_1(A)={\det }(A^{\mathcal {R}})\,\text{ and }\,{\mathrm{sdet}}_2(A)={\det }(A^{\mathcal {C}}). \end{aligned}$$

They have the following properties, which are almost the same as those of the normal determinant.

Lemma 3.1

Let \(A,C\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n},B\in {\mathbf {R}}^{n\times n},\lambda \in {{\mathbf {S}}}{{\mathbf {Q}}},w=1,2.\) Then

  1. (1).

    \({\mathrm{sdet}}_w(B)={\det }(B)^2\)

  2. (2).

    \({\mathrm{sdet}}_w(P(i,j))=1,{\mathrm{sdet}}_w(P(\lambda *i))={{\bar{\lambda }}} \lambda ,{\mathrm{sdet}}_w(P(i,\lambda *j))=1;\)

  3. (3).

    \({\mathrm{sdet}}_w(P(i,j)A)={\mathrm{sdet}}_w(A),{\mathrm{sdet}}_w(P(\lambda *i)A)={{\bar{\lambda }}} \lambda {\mathrm{sdet}}_w(A),\)

          \({\mathrm{sdet}}_w(P(i,\lambda *j)A)={\mathrm{sdet}}_w(A);\)

  4. (4).

    \({\mathrm{sdet}}_w(AC)={\mathrm{sdet}}_w(A){\mathrm{sdet}}_w(C)\);

  5. (5).

    \({\mathrm{sdet}}_w(A)={\bar{a}}_{11}a_{11}\cdots {\bar{a}}_{nn}a_{nn}\) if \(A=(a_{ij})\) is a triangle matrix.

By means of the elementary transformation, we can prove that the two determinants are the same.

Theorem 3.1

For \(A\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n},\) \({\mathrm{sdet}}_1(A)={\mathrm{sdet}}_2(A).\)

Proof

From Theorem 2.1, we know that there exist permutation matrices or unit lower triangular matrices \(P_1,P_2,\cdots , P_s\) and an upper triangular matrix U such that \(P_s\cdots P_2P_1A=U\triangleq (u_{ij}).\) From (2) and (3) of Lemma 3.1, we have

$$\begin{aligned} {\mathrm{sdet}}_1(A)={\mathrm{sdet}}_1(U)={\det }(U^{\mathcal {R}}),{\mathrm{sdet}}_2(A) ={\mathrm{sdet}}_2(U)={\det }(U^{\mathcal {C}}), \end{aligned}$$

where \(U^{\mathcal {R}}\) and \(U^{\mathcal {C}}\) are block triangular matrices with diagonal blocks \(u_{ii}^{\mathcal {R}}\) and \(u_{ii}^{\mathcal {C}}\). So, we can get

$$\begin{aligned} {\mathrm{sdet}}_1(A)=\prod _{i=1}^n{\det }(u_{ii}^{\mathcal {R}}) =\prod _{i=1}^n{\det }(u_{ii}^{\mathcal {C}})={\mathrm{sdet}}_2(A). \end{aligned}$$

\(\square \)

Remark 1

\(|A|_q={\mathrm{sdet}}_2(A)\). We omitted its proof.

Remark 2

In Theorem 2.1, U is not unique and the product of its diagonal elements is also not unique. But the modulus of the product is unique.

At the end of this subsection, we give the necessary and sufficient condition for the triangular decomposition of the split quaternion matrix.

Theorem 3.2

For \(A=(a_{ij}^{(1)})\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n}\), if \({\mathrm{sdet}}_1(A^{(k)})\not =0, k=1,2,\ldots ,n-1\), there exist an unit lower triangular matrix L and an upper triangular matrix U such that \(A=LU,\) where \(A^{(k)}=A(1:k,1:k)\) is the k-order leading principal submatrix of A.

Proof

Because \(|a_{11}^{(1)}|^2=|{\mathrm{sdet}}_1(A^{(1)})|\not =0,\) there exits

$$\begin{aligned} L_1=(0,-a_{21}^{(1)}(a_{11}^{(1)})^{-1}, \ldots ,-a_{n1}^{(1)}(a_{11}^{(1)})^{-1})^Te_1^T \end{aligned}$$

such that

$$\begin{aligned} L_1A=\left( \begin{array}{cccc}a_{11}^{(1)}&{}a_{12}^{(1)}&{}\cdots &{}a_{14}^{(1)}\\ 0&{}a_{22}^{(2)}&{}\cdots &{}a_{2n}^{(2)}\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ 0&{}a_{n2}^{(2)}&{}\cdots &{}a_{nn}^{(2)}\end{array}\right) . \end{aligned}$$
(3.1)

Partitioning \(L_1\) and A into

$$\begin{aligned} L_1=\left( \begin{array}{cc}L_{11}&{}0\\ L_{21}&{}L_{22}\end{array}\right) \,\text{ and }\,A=\left( \begin{array}{cc}A_{11}&{}A_{12}\\ A_{21}&{}A_{22}\end{array}\right) ,\end{aligned}$$
(3.2)

where \(L_{11}\) and \(A_{11}\) are 2-order matrices and \(L_{11}\) is unit low triangular, we have

$$\begin{aligned} L_{11}A_{11}=\left( \begin{array}{cc}a_{11}^{(1)}&{}a_{12}^{(1)}\\ 0&{}a_{22}^{(2)}\end{array}\right) . \end{aligned}$$

From

$$\begin{aligned} 0\not =|{\mathrm{sdet}}_1(A^{(2)})|=|{\mathrm{sdet}}_1(A_{11})|=|{\mathrm{sdet}}_1(L_{11}A_{11}) |=|a_{11}^{(1)}|^2|a_{22}^{(2)}|^2, \end{aligned}$$

it follows that \(|a_{22}^{(2)}|\not =0\) and Gaussian elimination process can continue. In the end, we can get

$$\begin{aligned} L_j=(\underbrace{0,\ldots ,0}_{j},-a_{j+1,j}^{(j)}(a_{jj}^{(j)})^{-1}, \ldots ,-a_{nj}^{(j)}(a_{jj}^{(j)})^{-1})^Te_j^T(j=1,2,\ldots ,n-1) \end{aligned}$$

with

$$\begin{aligned} L_{n-1}\cdots L_2L_1A=\left( \begin{array}{cccc}a_{11}^{(1)} &{}a_{12}^{(1)}&{}\cdots &{}a_{14}^{(1)}\\ 0&{}a_{22}^{(2)}&{}\cdots &{}a_{2n}^{(2)}\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ 0&{}0&{}\cdots &{}a_{nn}^{(n)}\end{array}\right) \triangleq U. \end{aligned}$$

Taking \(L=L_1^{-1}L_2^{-1}\cdots L_{n-1}^{-1}\), we have \(A=LU.\) \(\square \)

3.2 The Inverse

In this subsection, we give the definition of the inverse matrix and discuss its properties and computation method.

Naturally, we give the following definition.

Definition 2

Let \(A\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n}\). If there is a matrix \(B\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n}\) such that

$$\begin{aligned} AB=BA=I_n, \end{aligned}$$

we call A invertible and B is the inverse matrix of A.

In [2], the authors proved that \(AB=I\) is equivalent to \(BA=I\) and A is invertible if and only if \(\chi _A\) is invertible.

The following properties are basic and easy to be verified.

  1. (1).

    The diagonal split quaternion matrix \(A={\mathrm{diag}}(a_{11},\ldots ,a_{nn})\) is invertible if and only if \(|a_{ii}|\not =0,i=1,2,\ldots ,n.\) When A is invertible, \(A^{-1}={\mathrm{diag}}(a_{11}^{-1},\ldots ,a_{nn}^{-1}).\)

  2. (2).

    The block upper(low) triangular split quaternion matrix A is invertible if and only if \({\mathrm{sdet}}_1(A_{ii})\not =0,i=1,2,\cdots ,s.\) When A is invertible, \(A^{-1}\) is also block upper(low) triangular.

Theorem 3.3

Let \(A\in {{\mathbf {S}}}{{\mathbf {Q}}}^{n\times n}\). Then A is invertible if and only if \(A^{\mathcal {R}}\) is invertible.

Proof

If A is invertible, then there is B such that \(AB=BA=I\), which means \(A^{\mathcal {R}} B^{\mathcal {R}}=B^{\mathcal {R}} A^{\mathcal {R}}=I^{\mathcal {R}}=I_{2n}\), that is, \(A^{\mathcal {R}}\) is invertible.

On the contrary, If \(A^{\mathcal {R}}\) is invertible, then there is \(B\in {\mathbf {R}}^{2n\times 2n}\) such that \(A^{\mathcal {R}} B=BA^{\mathcal {R}}=I_{2n}\). Partitioned B into \(B=(B_{ij})\) with \(B_{ij}\in {\mathbf {R}}^{2\times 2},i,j=1,2,\cdots ,n\), then there is an unique set of numbers:\(B_{ij}^{(1)},B_{ij}^{(2)},B_{ij}^{(3)}\) and \(B_{ij}^{(4)}\) satisfying

$$\begin{aligned}&B_{ij}^{(1)}+B_{ij}^{(3)}=B_{ij}(1,1),-B_{ij}^{(2)}+B_{ij}^{(4)}=B_{ij}(1,2),\\&B_{ij}^{(2)}+B_{ij}^{(4)}=B_{ij}(2,1),B_{ij}^{(1)}-B_{ij}^{(3)}=B_{ij}(2,2). \end{aligned}$$

Taking \(\bar{b}_{ij}=B_{ij}^{(1)}+B_{ij}^{(2)}{\mathbf{i}}+B_{ij}^{(3)}{\mathbf{j}}+B_{ij}^{(4)}{\mathbf{k}}\) and \({\bar{B}}=({\bar{b}}_{ij})\), we have \({\bar{B}}^{\mathcal {R}}=B\) and \(A^{\mathcal {R}}{\bar{B}}^{\mathcal {R}}={\bar{B}}^{\mathcal {R}} A^{\mathcal {R}}=I_{2n},\) which means \(A{\bar{B}}={\bar{B}} A=I_n\), that is, A is invertible. \(\square \)

Corollary 3.1

A is invertible if and only if \({\mathrm{sdet}}_1(A)\not =0\).

Theorem 3.4

If A is an invertible split quaternion matrix, then there exits an invertible split quaternion matrix P such that \(PA=I\).

Proof

From Theorem 2.1, there exist permutation matrices or unit lower triangular matrices \(P_1,P_2,\ldots , P_s\) and an upper triangular matrix U such that \(P_s\cdots P_2P_1A=U\triangleq (u_{ij}).\)

It follows from \(0\not ={\mathrm{sdet}}_1(A)={\mathrm{sdet}}_1(U)\) that \(|u_{ii}|\not =0\) for all i. So there exist scaling elementary matrices or unit upper triangular matrices \(Q_1,Q_2,\ldots , Q_l\) such that

$$\begin{aligned} Q_l\cdots Q_2Q_1P_s\cdots P_2P_1A=Q_l\cdots Q_2Q_1 U=I. \end{aligned}$$

Taking \(P=Q_l\cdots Q_2Q_1P_s\cdots P_2P_1\), we complete the proof. \(\square \)

From the above theorem, we knows that, for the computation of the inverse of the split quaternion matrix, the method is the same as the real or complex matrix. Implementing three elementary row transformations on (A, E), we convert (A, E) to (E, B) and B is the inverse matrix of A.

4 Conclusions

In this paper, we discuss the elementary transformation for split quaternion matrices and give the process of triangulating a split matrix. Then we define two new determinants by means of real and complex representation matrices and obtain the sufficient condition for the existence of LU decomposition. Finally, we give the necessary and sufficient condition for the existence of the inverse matrix and derive the computation method of the inverse matrix by the elementary transformation.

The study on split quaternion matrices is not in-depth and comprehensive and we will continue to discuss its rank by the elementary transformation. Further, we will study linear equations by means of the rank and the determinant and so on.