Öz
In this study, Euler and De Moivre's formulas for fundamental matrices of commutative quaternions are obtained. Simple and effective methods are provided to find the powers and roots of these matrices with the aid of De Moivre's formula obtained from the fundamental matrices of commutative quaternions. Moreover, our results are supported by pseudo-codes and some examples.
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Anahtar Kelimeler
Commutative quaternions fundamental matrices Euler and De Moivre's formulas
Kaynakça
- Segre, C.: \emph{The real representations of complex elements and extension to bicomplex}. Systems. Math. Ann., 40, 413, (1892).
- Pei, S. C., Chang, J. H., Ding, J. J.: \emph{Commutative reduced biquaternions and their fourier transform for signal and image processing applications}. IEEE Trans. on Signal Proces., 52(7), 2012-2031, (2004).
- Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P.: \emph{The mathematics of minkowski space-time with an introduction to commutative hypercomplex numbers}. Birkhauser Verlag AG, Berlin, (2008).
- Pei S., Chang J., Ding J., Chen M.: \emph{Eigenvalues and singular value decompositions of reduced biquaternion matrices}. IEEE Trans. Circ. Syst. I., 55(9), 1549-8328, (2008).
- Isokawa, T., Nishimura, H., Matsui, N.: \emph{Commutative quaternion and multistate hopfield neural networks}. In Proc. Int. Joint Conf. Neural Netw., Barcelona, Spain, 1281-1286, (2010).
- Kosal H. H., Akyigit M., Tosun M.: \emph{Consimilarity of commutative quaternion matrices}. Miskolc Math. Notes, 16(2), 965-977, (2015).
- Yuan, S. F., Tian,Y., Li, M. Z.: \emph{On Hermitian solutions of the reduced biquaternion matrix equation $\left( AXB,\text{ }CXD \right)=\left( E,G \right)$}. Linear Multilinear Algebra, 1-19 (2018). DOI: 10.1080/03081087.2018.1543383.
- Kosal H. H., Tosun M.: \emph{Universal similarity factorization equalities for commutative quaternions and their matrices}. Linear Multilinear Algebra, 67(5), 926-938, (2019).
- Kosal H. H.: \emph{Least-squares solutions of the reduced biquaternion matrix equation $AX=B$ and their applications in colour image restoration.} J. Modern Opt., 66(18), 1802-1810, (2019).
- Zhang D., Guo Z., Wang G., Jiang T.: \emph{Algebraic techniques for least squares problems in commutative quaternionic theory}. Math Meth Appl Sci., 43, 3513-3523, (2020).
- Catoni, F., Cannata, R., Zampetti, P.: \emph{An introduction to commutative quaternions}. Adv. Appl. Clifford Algebras, 16, 1-28, (2006).
Öz
Kaynakça
- Segre, C.: \emph{The real representations of complex elements and extension to bicomplex}. Systems. Math. Ann., 40, 413, (1892).
- Pei, S. C., Chang, J. H., Ding, J. J.: \emph{Commutative reduced biquaternions and their fourier transform for signal and image processing applications}. IEEE Trans. on Signal Proces., 52(7), 2012-2031, (2004).
- Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., Zampetti, P.: \emph{The mathematics of minkowski space-time with an introduction to commutative hypercomplex numbers}. Birkhauser Verlag AG, Berlin, (2008).
- Pei S., Chang J., Ding J., Chen M.: \emph{Eigenvalues and singular value decompositions of reduced biquaternion matrices}. IEEE Trans. Circ. Syst. I., 55(9), 1549-8328, (2008).
- Isokawa, T., Nishimura, H., Matsui, N.: \emph{Commutative quaternion and multistate hopfield neural networks}. In Proc. Int. Joint Conf. Neural Netw., Barcelona, Spain, 1281-1286, (2010).
- Kosal H. H., Akyigit M., Tosun M.: \emph{Consimilarity of commutative quaternion matrices}. Miskolc Math. Notes, 16(2), 965-977, (2015).
- Yuan, S. F., Tian,Y., Li, M. Z.: \emph{On Hermitian solutions of the reduced biquaternion matrix equation $\left( AXB,\text{ }CXD \right)=\left( E,G \right)$}. Linear Multilinear Algebra, 1-19 (2018). DOI: 10.1080/03081087.2018.1543383.
- Kosal H. H., Tosun M.: \emph{Universal similarity factorization equalities for commutative quaternions and their matrices}. Linear Multilinear Algebra, 67(5), 926-938, (2019).
- Kosal H. H.: \emph{Least-squares solutions of the reduced biquaternion matrix equation $AX=B$ and their applications in colour image restoration.} J. Modern Opt., 66(18), 1802-1810, (2019).
- Zhang D., Guo Z., Wang G., Jiang T.: \emph{Algebraic techniques for least squares problems in commutative quaternionic theory}. Math Meth Appl Sci., 43, 3513-3523, (2020).
- Catoni, F., Cannata, R., Zampetti, P.: \emph{An introduction to commutative quaternions}. Adv. Appl. Clifford Algebras, 16, 1-28, (2006).
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 15 Ekim 2020 |
| Kabul Tarihi | 1 Ekim 2020 |
| Yayımlandığı Sayı | Yıl 2020 Cilt: 13 Sayı: 2 |
