Cayley–Hamilton theorem (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Cayley–Hamilton theorem" in English language version.

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Cayley 1889, pp. 475–496 Cayley, A. (1889). The Collected Mathematical Papers of Arthur Cayley. (Classic Reprint). Vol. 2. Forgotten books. ASIN B008HUED9O.

archive.org

Hamilton 1853, p. 562 Hamilton, W. R. (1853). Lectures on Quaternions. Dublin.{{cite book}}: CS1 maint: location missing publisher (link)

arxiv.org

See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients $c i$ is provided by Kondratyuk & Krivoruchenko (1992): $c_{i}=\sum _{k_{1},k_{2},\ldots ,k_{n}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} (A^{l})^{k_{l}},$ where the sum is taken over the sets of all integer partitions $k l \geq 0$ satisfying the equation $\sum _{l=1}^{n}lk_{l}=n-i.$ Krivoruchenko, M. I. (2016). "Trace Identities for Skew-Symmetric Matrices". arXiv:1605.00447 [math-ph]. Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group". Zeitschrift für Physik A. 344 (1): 99–115. Bibcode:1992ZPhyA.344...99K. doi:10.1007/BF01291027. S2CID 120467300.
Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see Alagös, Oral & Yüce (2012). The rings of quaternions and split-quaternions can both be represented by certain $2 \times 2$ complex matrices. (When restricted to unit norm, these are the groups $SU(2)$ and $SU(1,1)$ respectively.) Therefore it is not surprising that the theorem holds.
There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see Tian (2000). Alagös, Y.; Oral, K.; Yüce, S. (2012). "Split Quaternion Matrices". Miskolc Mathematical Notes. 13 (2): 223–232. doi:10.18514/MMN.2012.364. ISSN 1787-2405 (open access) Tian, Y. (2000). "Matrix representations of octonions and their application". Advances in Applied Clifford Algebras. 10 (1): 61–90. arXiv:math/0003166. Bibcode:2000math......3166T. CiteSeerX 10.1.1.237.2217. doi:10.1007/BF03042010. ISSN 0188-7009. S2CID 14465054.
Barut, Zeni & Laufer 1994a Barut, A. O.; Zeni, J. R.; Laufer, A. (1994a). "The exponential map for the conformal group O(2,4)". J. Phys. A: Math. Gen. 27 (15): 5239–5250. arXiv:hep-th/9408105. Bibcode:1994JPhA...27.5239B. doi:10.1088/0305-4470/27/15/022.
Barut, Zeni & Laufer 1994b Barut, A. O.; Zeni, J. R.; Laufer, A. (1994b). "The exponential map for the unitary group SU(2,2)". J. Phys. A: Math. Gen. 27 (20): 6799–6806. arXiv:hep-th/9408145. Bibcode:1994JPhA...27.6799B. doi:10.1088/0305-4470/27/20/017. S2CID 16495633.
Laufer 1997 Laufer, A. (1997). "The exponential map of GL(N)". J. Phys. A: Math. Gen. 30 (15): 5455–5470. arXiv:hep-th/9604049. Bibcode:1997JPhA...30.5455L. doi:10.1088/0305-4470/30/15/029. S2CID 10699434.
Curtright, Fairlie & Zachos 2014 Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10 (2014): 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084. S2CID 18776942.

doi.org

See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients $c i$ is provided by Kondratyuk & Krivoruchenko (1992): $c_{i}=\sum _{k_{1},k_{2},\ldots ,k_{n}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} (A^{l})^{k_{l}},$ where the sum is taken over the sets of all integer partitions $k l \geq 0$ satisfying the equation $\sum _{l=1}^{n}lk_{l}=n-i.$ Krivoruchenko, M. I. (2016). "Trace Identities for Skew-Symmetric Matrices". arXiv:1605.00447 [math-ph]. Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group". Zeitschrift für Physik A. 344 (1): 99–115. Bibcode:1992ZPhyA.344...99K. doi:10.1007/BF01291027. S2CID 120467300.
See, e.g., p. 54 of Brown 1994, which solves Jacobi's formula, ${\frac {\partial p(\lambda )}{\partial \lambda }}=p(\lambda )\sum _{m=0}^{\infty }\lambda ^{-(m+1)}\operatorname {tr} A^{m}=p(\lambda )~\operatorname {tr} {\frac {I}{\lambda I-A}}\equiv \operatorname {tr} B~,$ where $B$ is the adjugate matrix of the next section. There also exists an equivalent, related recursive algorithm introduced by Urbain Le Verrier and Dmitry Konstantinovich Faddeev—the Faddeev–LeVerrier algorithm, which reads ${\begin{aligned}M_{0}&\equiv O&c_{n}&=1\qquad &(k=0)\\[5pt]M_{k}&\equiv AM_{k-1}-{\frac {1}{k-1}}(\operatorname {tr} (AM_{k-1}))I\qquad \qquad &c_{n-k}&=-{\frac {1}{k}}\operatorname {tr} (AM_{k})\qquad &k=1,\ldots ,n~.\end{aligned}}$ (see, e.g., Gantmacher 1960, p. 88.) Observe $A -1 = - M n / c 0$ as the recursion terminates. See the algebraic proof in the following section, which relies on the modes of the adjugate, $B k \equiv M n - k$ . Specifically, $(\lambda I-A)B=Ip(\lambda )$ and the above derivative of $p$ when one traces it yields $\lambda p'-np=\operatorname {tr} (AB)~,$ (Hou 1998), and the above recursions, in turn. Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3. Gantmacher, F.R. (1960). The Theory of Matrices. NY: Chelsea Publishing. ISBN 978-0-8218-1376-8. {{cite book}}: ISBN / Date incompatibility (help) Hou, S. H. (1998). "Classroom Note: A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm". SIAM Review. 40 (3): 706–709. Bibcode:1998SIAMR..40..706H. doi:10.1137/S003614459732076X. "Classroom Note: A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm"
Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see Alagös, Oral & Yüce (2012). The rings of quaternions and split-quaternions can both be represented by certain $2 \times 2$ complex matrices. (When restricted to unit norm, these are the groups $SU(2)$ and $SU(1,1)$ respectively.) Therefore it is not surprising that the theorem holds.
There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see Tian (2000). Alagös, Y.; Oral, K.; Yüce, S. (2012). "Split Quaternion Matrices". Miskolc Mathematical Notes. 13 (2): 223–232. doi:10.18514/MMN.2012.364. ISSN 1787-2405 (open access) Tian, Y. (2000). "Matrix representations of octonions and their application". Advances in Applied Clifford Algebras. 10 (1): 61–90. arXiv:math/0003166. Bibcode:2000math......3166T. CiteSeerX 10.1.1.237.2217. doi:10.1007/BF03042010. ISSN 0188-7009. S2CID 14465054.
Crilly 1998 Crilly, T. (1998). "The young Arthur Cayley". Notes Rec. R. Soc. Lond. 52 (2): 267–282. doi:10.1098/rsnr.1998.0050. S2CID 146669911.
Frobenius 1878 Frobenius, G. (1878). "Ueber lineare Substutionen und bilineare Formen". J. Reine Angew. Math. 1878 (84): 1–63. doi:10.1515/crll.1878.84.1 (inactive 2 January 2025).{{cite journal}}: CS1 maint: DOI inactive as of January 2025 (link)
Zeni & Rodrigues 1992 Zeni, J. R.; Rodrigues, W.A. (1992). "A thoughtful study of Lorentz transformations by Clifford algebras". Int. J. Mod. Phys. A. 7 (8): 1793 pp. Bibcode:1992IJMPA...7.1793Z. doi:10.1142/S0217751X92000776.
Barut, Zeni & Laufer 1994a Barut, A. O.; Zeni, J. R.; Laufer, A. (1994a). "The exponential map for the conformal group O(2,4)". J. Phys. A: Math. Gen. 27 (15): 5239–5250. arXiv:hep-th/9408105. Bibcode:1994JPhA...27.5239B. doi:10.1088/0305-4470/27/15/022.
Barut, Zeni & Laufer 1994b Barut, A. O.; Zeni, J. R.; Laufer, A. (1994b). "The exponential map for the unitary group SU(2,2)". J. Phys. A: Math. Gen. 27 (20): 6799–6806. arXiv:hep-th/9408145. Bibcode:1994JPhA...27.6799B. doi:10.1088/0305-4470/27/20/017. S2CID 16495633.
Laufer 1997 Laufer, A. (1997). "The exponential map of GL(N)". J. Phys. A: Math. Gen. 30 (15): 5455–5470. arXiv:hep-th/9604049. Bibcode:1997JPhA...30.5455L. doi:10.1088/0305-4470/30/15/029. S2CID 10699434.
Curtright, Fairlie & Zachos 2014 Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10 (2014): 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084. S2CID 18776942.
Straubing, Howard (1983-01-01). "A combinatorial proof of the Cayley-Hamilton theorem". Discrete Mathematics. 43 (2): 273–279. doi:10.1016/0012-365X(83)90164-4. ISSN 0012-365X.
Zhang 1997 Zhang, F. (1997). "Quaternions and matrices of quaternions". Linear Algebra and Its Applications. 251: 21–57. doi:10.1016/0024-3795(95)00543-9. ISSN 0024-3795 (open archive).

elsevier.com

linkinghub.elsevier.com

Straubing, Howard (1983-01-01). "A combinatorial proof of the Cayley-Hamilton theorem". Discrete Mathematics. 43 (2): 273–279. doi:10.1016/0012-365X(83)90164-4. ISSN 0012-365X.

harvard.edu

ui.adsabs.harvard.edu

See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients $c i$ is provided by Kondratyuk & Krivoruchenko (1992): $c_{i}=\sum _{k_{1},k_{2},\ldots ,k_{n}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} (A^{l})^{k_{l}},$ where the sum is taken over the sets of all integer partitions $k l \geq 0$ satisfying the equation $\sum _{l=1}^{n}lk_{l}=n-i.$ Krivoruchenko, M. I. (2016). "Trace Identities for Skew-Symmetric Matrices". arXiv:1605.00447 [math-ph]. Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group". Zeitschrift für Physik A. 344 (1): 99–115. Bibcode:1992ZPhyA.344...99K. doi:10.1007/BF01291027. S2CID 120467300.
See, e.g., p. 54 of Brown 1994, which solves Jacobi's formula, ${\frac {\partial p(\lambda )}{\partial \lambda }}=p(\lambda )\sum _{m=0}^{\infty }\lambda ^{-(m+1)}\operatorname {tr} A^{m}=p(\lambda )~\operatorname {tr} {\frac {I}{\lambda I-A}}\equiv \operatorname {tr} B~,$ where $B$ is the adjugate matrix of the next section. There also exists an equivalent, related recursive algorithm introduced by Urbain Le Verrier and Dmitry Konstantinovich Faddeev—the Faddeev–LeVerrier algorithm, which reads ${\begin{aligned}M_{0}&\equiv O&c_{n}&=1\qquad &(k=0)\\[5pt]M_{k}&\equiv AM_{k-1}-{\frac {1}{k-1}}(\operatorname {tr} (AM_{k-1}))I\qquad \qquad &c_{n-k}&=-{\frac {1}{k}}\operatorname {tr} (AM_{k})\qquad &k=1,\ldots ,n~.\end{aligned}}$ (see, e.g., Gantmacher 1960, p. 88.) Observe $A -1 = - M n / c 0$ as the recursion terminates. See the algebraic proof in the following section, which relies on the modes of the adjugate, $B k \equiv M n - k$ . Specifically, $(\lambda I-A)B=Ip(\lambda )$ and the above derivative of $p$ when one traces it yields $\lambda p'-np=\operatorname {tr} (AB)~,$ (Hou 1998), and the above recursions, in turn. Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3. Gantmacher, F.R. (1960). The Theory of Matrices. NY: Chelsea Publishing. ISBN 978-0-8218-1376-8. {{cite book}}: ISBN / Date incompatibility (help) Hou, S. H. (1998). "Classroom Note: A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm". SIAM Review. 40 (3): 706–709. Bibcode:1998SIAMR..40..706H. doi:10.1137/S003614459732076X. "Classroom Note: A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm"
Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see Alagös, Oral & Yüce (2012). The rings of quaternions and split-quaternions can both be represented by certain $2 \times 2$ complex matrices. (When restricted to unit norm, these are the groups $SU(2)$ and $SU(1,1)$ respectively.) Therefore it is not surprising that the theorem holds.
There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see Tian (2000). Alagös, Y.; Oral, K.; Yüce, S. (2012). "Split Quaternion Matrices". Miskolc Mathematical Notes. 13 (2): 223–232. doi:10.18514/MMN.2012.364. ISSN 1787-2405 (open access) Tian, Y. (2000). "Matrix representations of octonions and their application". Advances in Applied Clifford Algebras. 10 (1): 61–90. arXiv:math/0003166. Bibcode:2000math......3166T. CiteSeerX 10.1.1.237.2217. doi:10.1007/BF03042010. ISSN 0188-7009. S2CID 14465054.
Zeni & Rodrigues 1992 Zeni, J. R.; Rodrigues, W.A. (1992). "A thoughtful study of Lorentz transformations by Clifford algebras". Int. J. Mod. Phys. A. 7 (8): 1793 pp. Bibcode:1992IJMPA...7.1793Z. doi:10.1142/S0217751X92000776.
Barut, Zeni & Laufer 1994a Barut, A. O.; Zeni, J. R.; Laufer, A. (1994a). "The exponential map for the conformal group O(2,4)". J. Phys. A: Math. Gen. 27 (15): 5239–5250. arXiv:hep-th/9408105. Bibcode:1994JPhA...27.5239B. doi:10.1088/0305-4470/27/15/022.
Barut, Zeni & Laufer 1994b Barut, A. O.; Zeni, J. R.; Laufer, A. (1994b). "The exponential map for the unitary group SU(2,2)". J. Phys. A: Math. Gen. 27 (20): 6799–6806. arXiv:hep-th/9408145. Bibcode:1994JPhA...27.6799B. doi:10.1088/0305-4470/27/20/017. S2CID 16495633.
Laufer 1997 Laufer, A. (1997). "The exponential map of GL(N)". J. Phys. A: Math. Gen. 30 (15): 5455–5470. arXiv:hep-th/9604049. Bibcode:1997JPhA...30.5455L. doi:10.1088/0305-4470/30/15/029. S2CID 10699434.
Curtright, Fairlie & Zachos 2014 Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10 (2014): 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084. S2CID 18776942.

psu.edu

citeseerx.ist.psu.edu

Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see Alagös, Oral & Yüce (2012). The rings of quaternions and split-quaternions can both be represented by certain $2 \times 2$ complex matrices. (When restricted to unit norm, these are the groups $SU(2)$ and $SU(1,1)$ respectively.) Therefore it is not surprising that the theorem holds.
There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see Tian (2000). Alagös, Y.; Oral, K.; Yüce, S. (2012). "Split Quaternion Matrices". Miskolc Mathematical Notes. 13 (2): 223–232. doi:10.18514/MMN.2012.364. ISSN 1787-2405 (open access) Tian, Y. (2000). "Matrix representations of octonions and their application". Advances in Applied Clifford Algebras. 10 (1): 61–90. arXiv:math/0003166. Bibcode:2000math......3166T. CiteSeerX 10.1.1.237.2217. doi:10.1007/BF03042010. ISSN 0188-7009. S2CID 14465054.

semanticscholar.org

api.semanticscholar.org

See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients $c i$ is provided by Kondratyuk & Krivoruchenko (1992): $c_{i}=\sum _{k_{1},k_{2},\ldots ,k_{n}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} (A^{l})^{k_{l}},$ where the sum is taken over the sets of all integer partitions $k l \geq 0$ satisfying the equation $\sum _{l=1}^{n}lk_{l}=n-i.$ Krivoruchenko, M. I. (2016). "Trace Identities for Skew-Symmetric Matrices". arXiv:1605.00447 [math-ph]. Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group". Zeitschrift für Physik A. 344 (1): 99–115. Bibcode:1992ZPhyA.344...99K. doi:10.1007/BF01291027. S2CID 120467300.
Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see Alagös, Oral & Yüce (2012). The rings of quaternions and split-quaternions can both be represented by certain $2 \times 2$ complex matrices. (When restricted to unit norm, these are the groups $SU(2)$ and $SU(1,1)$ respectively.) Therefore it is not surprising that the theorem holds.
There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see Tian (2000). Alagös, Y.; Oral, K.; Yüce, S. (2012). "Split Quaternion Matrices". Miskolc Mathematical Notes. 13 (2): 223–232. doi:10.18514/MMN.2012.364. ISSN 1787-2405 (open access) Tian, Y. (2000). "Matrix representations of octonions and their application". Advances in Applied Clifford Algebras. 10 (1): 61–90. arXiv:math/0003166. Bibcode:2000math......3166T. CiteSeerX 10.1.1.237.2217. doi:10.1007/BF03042010. ISSN 0188-7009. S2CID 14465054.
Crilly 1998 Crilly, T. (1998). "The young Arthur Cayley". Notes Rec. R. Soc. Lond. 52 (2): 267–282. doi:10.1098/rsnr.1998.0050. S2CID 146669911.
Barut, Zeni & Laufer 1994b Barut, A. O.; Zeni, J. R.; Laufer, A. (1994b). "The exponential map for the unitary group SU(2,2)". J. Phys. A: Math. Gen. 27 (20): 6799–6806. arXiv:hep-th/9408145. Bibcode:1994JPhA...27.6799B. doi:10.1088/0305-4470/27/20/017. S2CID 16495633.
Laufer 1997 Laufer, A. (1997). "The exponential map of GL(N)". J. Phys. A: Math. Gen. 30 (15): 5455–5470. arXiv:hep-th/9604049. Bibcode:1997JPhA...30.5455L. doi:10.1088/0305-4470/30/15/029. S2CID 10699434.
Curtright, Fairlie & Zachos 2014 Curtright, T L; Fairlie, D B; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA. 10 (2014): 084. arXiv:1402.3541. Bibcode:2014SIGMA..10..084C. doi:10.3842/SIGMA.2014.084. S2CID 18776942.

siam.org

epubs.siam.org

See, e.g., p. 54 of Brown 1994, which solves Jacobi's formula, ${\frac {\partial p(\lambda )}{\partial \lambda }}=p(\lambda )\sum _{m=0}^{\infty }\lambda ^{-(m+1)}\operatorname {tr} A^{m}=p(\lambda )~\operatorname {tr} {\frac {I}{\lambda I-A}}\equiv \operatorname {tr} B~,$ where $B$ is the adjugate matrix of the next section. There also exists an equivalent, related recursive algorithm introduced by Urbain Le Verrier and Dmitry Konstantinovich Faddeev—the Faddeev–LeVerrier algorithm, which reads ${\begin{aligned}M_{0}&\equiv O&c_{n}&=1\qquad &(k=0)\\[5pt]M_{k}&\equiv AM_{k-1}-{\frac {1}{k-1}}(\operatorname {tr} (AM_{k-1}))I\qquad \qquad &c_{n-k}&=-{\frac {1}{k}}\operatorname {tr} (AM_{k})\qquad &k=1,\ldots ,n~.\end{aligned}}$ (see, e.g., Gantmacher 1960, p. 88.) Observe $A -1 = - M n / c 0$ as the recursion terminates. See the algebraic proof in the following section, which relies on the modes of the adjugate, $B k \equiv M n - k$ . Specifically, $(\lambda I-A)B=Ip(\lambda )$ and the above derivative of $p$ when one traces it yields $\lambda p'-np=\operatorname {tr} (AB)~,$ (Hou 1998), and the above recursions, in turn. Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3. Gantmacher, F.R. (1960). The Theory of Matrices. NY: Chelsea Publishing. ISBN 978-0-8218-1376-8. {{cite book}}: ISBN / Date incompatibility (help) Hou, S. H. (1998). "Classroom Note: A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm". SIAM Review. 40 (3): 706–709. Bibcode:1998SIAMR..40..706H. doi:10.1137/S003614459732076X. "Classroom Note: A Simple Proof of the Leverrier--Faddeev Characteristic Polynomial Algorithm"

uni-jena.de

zs.thulb.uni-jena.de

Hamilton 1862 Hamilton, W. R. (1862). "On the Existence of a Symbolic and Biquadratic Equation which is satisfied by the Symbol of Linear or Distributive Operation on a Quaternion". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. series iv. 24: 127–128. ISSN 1478-6435. Retrieved 2015-02-14.

worldcat.org

search.worldcat.org

Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see Alagös, Oral & Yüce (2012). The rings of quaternions and split-quaternions can both be represented by certain $2 \times 2$ complex matrices. (When restricted to unit norm, these are the groups $SU(2)$ and $SU(1,1)$ respectively.) Therefore it is not surprising that the theorem holds.
There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see Tian (2000). Alagös, Y.; Oral, K.; Yüce, S. (2012). "Split Quaternion Matrices". Miskolc Mathematical Notes. 13 (2): 223–232. doi:10.18514/MMN.2012.364. ISSN 1787-2405 (open access) Tian, Y. (2000). "Matrix representations of octonions and their application". Advances in Applied Clifford Algebras. 10 (1): 61–90. arXiv:math/0003166. Bibcode:2000math......3166T. CiteSeerX 10.1.1.237.2217. doi:10.1007/BF03042010. ISSN 0188-7009. S2CID 14465054.
Hamilton 1862 Hamilton, W. R. (1862). "On the Existence of a Symbolic and Biquadratic Equation which is satisfied by the Symbol of Linear or Distributive Operation on a Quaternion". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. series iv. 24: 127–128. ISSN 1478-6435. Retrieved 2015-02-14.
Straubing, Howard (1983-01-01). "A combinatorial proof of the Cayley-Hamilton theorem". Discrete Mathematics. 43 (2): 273–279. doi:10.1016/0012-365X(83)90164-4. ISSN 0012-365X.
Zhang 1997 Zhang, F. (1997). "Quaternions and matrices of quaternions". Linear Algebra and Its Applications. 251: 21–57. doi:10.1016/0024-3795(95)00543-9. ISSN 0024-3795 (open archive).

wstein.org

Stein, William. Algebraic Number Theory, a Computational Approach (PDF). p. 29.