Cayley–Hamilton theorem (English Wikipedia)
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- See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients ci is provided by Kondratyuk & Krivoruchenko (1992): where the sum is taken over the sets of all integer partitions kl ≥ 0 satisfying the equation 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,
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
(see, e.g., Gantmacher 1960, p. 88.) Observe A−1 = −Mn /c0 as the recursion terminates.
See the algebraic proof in the following section, which relies on the modes of the adjugate, Bk ≡ Mn−k.
Specifically, and the above derivative of p when one traces it yields
(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 × 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.{{cite journal}}: Cite uses deprecated parameter|citeseerx=(help) - Cayley 1858, pp. 17–37 Cayley, A. (1858). "A Memoir on the Theory of Matrices". Philos. Trans. 148: 17. Bibcode:1858RSPT..148...17C.
- Cayley 1858, p. 24. Cayley, A. (1858). "A Memoir on the Theory of Matrices". Philos. Trans. 148: 17. Bibcode:1858RSPT..148...17C.
- 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.
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14ee340d1fd2cbc359cd844e75cc1bb8b3284a3b)
amazon.com (Global: 159th place; English: 105th place)
- 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 (Global: 6th place; English: 6th place)
- Hamilton 1853, p. 562 Hamilton, W. R. (1853). Lectures on Quaternions. Dublin.
{{cite book}}: CS1 maint: location missing publisher (link)
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- See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients ci is provided by Kondratyuk & Krivoruchenko (1992): where the sum is taken over the sets of all integer partitions kl ≥ 0 satisfying the equation 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 × 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.{{cite journal}}: Cite uses deprecated parameter|citeseerx=(help) - 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.
degruyterbrill.com (Global: 6,220th place; English: 3,995th place)
- Frobenius 1878 Frobenius, G. (1878). "Ueber lineare Substutionen und bilineare Formen". J. Reine Angew. Math. 1878 (84): 1–63.
doi.org (Global: 2nd place; English: 2nd place)
- See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients ci is provided by Kondratyuk & Krivoruchenko (1992): where the sum is taken over the sets of all integer partitions kl ≥ 0 satisfying the equation 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,
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
(see, e.g., Gantmacher 1960, p. 88.) Observe A−1 = −Mn /c0 as the recursion terminates.
See the algebraic proof in the following section, which relies on the modes of the adjugate, Bk ≡ Mn−k.
Specifically, and the above derivative of p when one traces it yields
(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 × 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.{{cite journal}}: Cite uses deprecated parameter|citeseerx=(help) - 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.
- 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 (Global: 365th place; English: 372nd place)
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- 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.
psu.edu (Global: 299th place; English: 159th place)
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- 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 × 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.{{cite journal}}: Cite uses deprecated parameter|citeseerx=(help)
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- See Sect. 2 of Krivoruchenko (2016). An explicit expression for the coefficients ci is provided by Kondratyuk & Krivoruchenko (1992): where the sum is taken over the sets of all integer partitions kl ≥ 0 satisfying the equation 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 × 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.{{cite journal}}: Cite uses deprecated parameter|citeseerx=(help) - 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.
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- See, e.g., p. 54 of Brown 1994, which solves Jacobi's formula,
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
(see, e.g., Gantmacher 1960, p. 88.) Observe A−1 = −Mn /c0 as the recursion terminates.
See the algebraic proof in the following section, which relies on the modes of the adjugate, Bk ≡ Mn−k.
Specifically, and the above derivative of p when one traces it yields
(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"
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- 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.
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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 × 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.{{cite journal}}: Cite uses deprecated parameter|citeseerx=(help) - 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 (Global: low place; English: low place)
- Stein, William. Algebraic Number Theory, a Computational Approach (PDF). p. 29.