Split quaternion algebra is not a Euclidean distance space because of having zero divisors. Thus, the traditional QR decomposition based on Givens rotations and Householder reflection transformations is difficult to implement. To overcome this difficulty and to address the non-commutativity of split quaternion multiplication, we utilize the real representation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A^\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mi>σ</mi> </msup> </math></EquationSource> </InlineEquation> of the split quaternion matrix A. By leveraging the proposed decomposition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A^\sigma = \widetilde{Q}R_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>σ</mi> </msup> <mo>=</mo> <mover accent="true"> <mi>Q</mi> <mo stretchy="true">~</mo> </mover> <msub> <mi>R</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\widetilde{Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>Q</mi> <mo stretchy="true">~</mo> </mover> </math></EquationSource> </InlineEquation> is an orthogonal matrix, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R_4 = \begin{bmatrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{bmatrix}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>4</mn> </msub> <mo>=</mo> <mfenced close="]" open="["> <mrow> <mtable> <mtr> <mtd> <msub> <mi>R</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>R</mi> <mn>12</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <msub> <mi>R</mi> <mn>21</mn> </msub> </mrow> </mtd> <mtd> <msub> <mi>R</mi> <mn>22</mn> </msub> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_{11}, R_{12}, R_{21}, R_{22}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>11</mn> </msub> <mo>,</mo> <msub> <mi>R</mi> <mn>12</mn> </msub> <mo>,</mo> <msub> <mi>R</mi> <mn>21</mn> </msub> <mo>,</mo> <msub> <mi>R</mi> <mn>22</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> being upper triangular), the QR decomposition of A is successfully constructed and the corresponding algorithms are developed.The experimental results show that it performs well in both speed and accuracy.

An algorithm for QR decomposition of split quaternion matrices

期刊论文
发表日期：

2025年12月15日

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Qianqian Liu,
Xin Liu,
Jianhai Lin,
Yang Zhang

摘要

Split quaternion algebra is not a Euclidean distance space because of having zero divisors. Thus, the traditional QR decomposition based on Givens rotations and Householder reflection transformations is difficult to implement. To overcome this difficulty and to address the non-commutativity of split quaternion multiplication, we utilize the real representation $A^\sigma $ $A^{σ}$ of the split quaternion matrix A. By leveraging the proposed decomposition $A^\sigma = \widetilde{Q}R_4$ $A^{σ} = \tilde{Q} R_{4}$ ( $\widetilde{Q}$ $\tilde{Q}$ is an orthogonal matrix, and $R_4 = \begin{bmatrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{bmatrix}$ $R_{4} = [\begin{matrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{matrix}]$ with $R_{11}, R_{12}, R_{21}, R_{22}$ $R_{11}, R_{12}, R_{21}, R_{22}$ being upper triangular), the QR decomposition of A is successfully constructed and the corresponding algorithms are developed.The experimental results show that it performs well in both speed and accuracy.