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

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Tatiana Klimchuk, Vladimir V. Sergeichuk

Representation Theory · Submitted 8 Dec 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1412.2801 (version 1)
Pith IDPR66J2P3, Pith's permanent record of this paper and its review
CitationTatiana Klimchuk, Vladimir V. Sergeichuk. "Consimilarity and quaternion matrix equations $AX-\hat{X}B=C$, $X-A\hat{X}B=C$." arXiv:1412.2801 (2014). https://pith.science/paper/PR66J2P3
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,236 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.84 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 $X-A\hat{X}B=C$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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 $h=a+bi+cj+dk$. background 1.00 0.03 0.05 asserted 1.00
s1 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. contribution 0.93 0.96 0.49 asserted 1.00
s2 We apply the obtained canonical form to the quaternion matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$. contribution 0.92 0.94 0.29 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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 $X-A\hat{X}B=C$.

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