The Real Representation of Canonical Hyperbolic Quaternion Matrices and Its Applications

In this paper, we construct the real representation matrix of canonical hyperbolic quaternion matrices and give some properties in detail. Then, by means of the real representation, we study linear equations, the inverse and the generalized inverse of the canonical hyperbolic quaternion matrix and get some interesting results.


Introduction
In 1843, Hamilton introduced the quaternion, which has the form of , , , a a a a are real numbers.Since quaternions are non-commutative, they differ from complex numbers and real numbers.Quaternions and quaternion matrices play an important role in quaternionic quantum mechanics and field theory [1].In 1849, the split quaternion(or coquaternion), which was found by James Cockle, is in the form of , , , a a a a are real numbers.Split quaternions are noncommutative, too.But split quaternion set contains zero-divisors, nilpotent elements and nontrivial idempotents [2,3].
In 1892, Segre proposed modified quaternions so that commutative property in multiplication is possible [4].In Catoni, et al. [5], the authors studied three three types of commutative quaternions: Elliptic quaternions, Parabolic quaternions and Hyperbolic quaternions.They are 4-dimensional like the set of quaternions, but contain zero-divisor and isotropic elements.Although commutative quaternion algebra theory is becoming more and more important in recent years and has many important applications in the areas of mathematics and physics [5][6][7][8][9][10], the current focus is mainly on canonical elliptic quaternions [11][12][13][14].In these papers, H. Kösal and M. Tosun gave some properties of canonical elliptic quaternions and their fundamental matrices.After that, they investigated canonical elliptic quaternion matrices using properties of complex matrices.Then they defined the complex adjoint matrix(complex representation matrix) of canonical elliptic quaternion matrices and gave some of their properties.Recently, they proposed real matrix representations of canonical elliptic quaternions and their matrices and derived their algebraic properties and fundamental equations.
As has been noticed, there is no paper that studied the theory on canonical hyperbolic quaternion matrices.In this paper, we will discuss canonical hyperbolic quaternion matrices.
Let R denote the real number field and = This paper is organized as follows.In Section 2, we construct the real representation of canonical hyperbolic quaternion matrices and systematically study its properties.In Section 3, we discuss the canonical hyperbolic quaternion linear equations and study the judgment and construction of solutions.Next, we give the necessary and sufficient condition for canonical hyperbolic quaternion matrix invertibility.Finally, we define a generalized inverse and initially discuss its existence and uniqueness.Some results are interesting.In Section 4, we give some conclusions.

Real Representation of Canonical Hyperbolic Quaternion Matrices
In this section, we define the real representation of canonical hyperbolic quaternion matrices and systematically study its properties.It is worth mentioning that, unlike other quaternions, the canonical hyperbolic quaternion is not the natural generalization of complex number.It is hard for its matrices to construct the complex representation and we only discuss the real representation.For real representations of quaternion matrices, split quaternion matrices and elliptic quaternion matrices, many results have been obtained [2,3,[14][15][16][17] and their references for details).Inspired by them, we define the real representation of canonical hyperbolic quaternion matrices as follows.
For any we define its real representation matrix or real representation R A as follows. .

R m n A A A A A A A A A A A A A
1) The set of all matrices shaped like (2.1) is denoted by By simple computation, we can obtain the following properties.

Theorem
Let , , , It is easy to verify that the following results are right.

Theorem
For any  Rr and it is the real representation matrix of the canonical hyperbolic quaternion matrix and V is the real representation matrix of the canonical hyperbolic quaternion matrix W Further, we can also get the following construction method. .A is the transpose of A .For the above concepts, the following results can be easily verified.
U is a orthogonal matrix, and vice versa.

Some Applications of the Real Representation
Various quaternion matrix equations have been studied in a large number of papers.In He, et al. [18], Structure [19], He, et al. [20], the authors discussed some quaternion matrix equation and equations by means of matrix decomposition, rank equality, real representation and so on.In Zhang, et al. [2], Jiang, et al. [21], the authors studied some split quaternion matrix equations by real or complex representation.In Kösal and Tosun [14], the authors proposed the real matrix representation of canonical elliptic quaternion matrices and considered their equations.In this section, we study some applications of the real representation, including linear equation, inverse and MP inverse.
A S y S A S S y S b m and then ( , , , ) From Theorem 2.3, we know that ( , , , ) n n n y Q y R y S y is the real representation matrix of a canonical hyperbolic quaternion matrix(marked as x ), and obtain = Ax b .. In conclusion, we have the following result.

Let
,.For the inverse, we have the following result.

Theorem
Let .

Rr
. And so we have By the way, we can get the following interesting conclusion.

Rr
, that is, D and its inverse have the same structure.
By summing up the above conclusions, we can naturally get the following result.
A is invertible; (4) .( ) = 4 R rank A n .By similar derivation, we can obtain the following further conclusions.

Theorem
A YB C has the unique solution Y , then Next, we define a generalized inverse.Let .From the above discussion, we can get the following conclusion.

Theorem
Let .Then D and its MP inverse have the same structure.
Fianlly, we give an example.Let

Rr
The T -MP inverse of A is 1

Conclusions
In this paper, we construct the real representation of canonical hyperbolic quaternion matrices and systematically study its properties.Then, we discuss the canonical hyperbolic quaternion linear equations and study the judgment and construction of solutions.Next, we give the necessary and sufficient condition for canonical hyperbolic quaternion matrix invertibility.Finally, we define a generalized inverse and initially discuss its existence and uniqueness.Some results are interesting.
We have only initially studied canonical hyperbolic quaternion matrices, and there are still a lot of problems worthy of further discussion.For example, rank, norm, determinant, etc.In the future work, we will pay more attention to the least squares problem.
R denote the canonical hyperbolic quaternion set, = = , = = , = = .i j k ij ji k jk kj i ki ik j a a a i a j a k b b b i b j b k . U is a orthogonal matrix if and only if R U is a orthogonal matrix.Proof.(1)and (2) can be easily verified.
) If U is a orthogonal matrix, i.e., = T UU I .By (1) of Theorem 2.1 and (2) of this theorem, we have 4 , where the symbol ( : , : ) M i j k l represents the submatrix of M containing the intersection of rows i to j and columns k to l .On the other hand, if (3.2) has a solution n y  R , then we have = = (1: 4 ,1),

Q
if and only if the real linear equation (3.2) has a solution in 4n R .And if real linear equation (3.2) has the solution 4n I j I k y is the solution of (3.1).
inverse (if exists) also belongs to 44 nn 

QQ
Then the following are equivalent.(1).= Ax b has a unique solution; we call X as T Moore-Penrose( T -MP) inverse of A .Let Y is the MP inverse of R A , that is, Y satisfies the conditions ( . A has a unique T -MP inverse.T -MP inverse of A .By the way, we can also get the following interesting conclusion. 

Rr 2.4. Theorem.
invertible if and only if RR AA Proof.If A is invertible, there exists nn B . Through the above proof and the uniqueness of the inverse matrix, we can know that the inverse B of For the same reason, nm S YS and nm R YR are both MP inverses of