Abstract
The minimum equation. If A is a matrix of order n over a field p, the matrices I, A, A 2,..., A n 2 constitute n 2 + 1 sets of n 2 numbers each, and hence are linearly dependent in p. Thus A satisfies some equation
with coefficients in p of minimum degree μ. We shall call μ the index of A. The index of a scalar matrix is 1. Every matrix except 0 has an index.
Keywords
- Characteristic Equation
- Characteristic Root
- Hermitian Matrix
- Irreducible Factor
- Elementary Symmetric Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Notes
Frobenius: J. reine angew. Math. Vol. 84 (1878) p. 1–63.
Phillips, H. B.: Amer. J. Math. Vol. 41 (1919) pp. 266–278.
Pasch: Math. Ann. Vol. 38 (1891) pp. 24–49.
Cayley, A.: Philos. Trans. Roy. Soc, London Vol. 148 (1858) pp. 17–37.
Laguerre, E.: J. École polytechn. Vol. 25 (1867) pp. 215–264.
Forsyth, A. R.: Mess. Math. Vol. 13 (1884) pp. 139–142.
Frobenius: Mess. Math. Vol. 13 (1884) pp. 62–66.
Buchheim: Proc. London Math. Soc. Vol. 16 (1884) pp. 63–82.
Cauchy: Exercises d’analyse et de physique mathématique Vol. 1 (1840) p. 53.
Günther, S.: Z. Math. Vol. 21 (1876) pp. 178–191.
Laisant, G. A.: Bull. SOC, Math. France Vol. 17 (1889) pp. 104–107.
Rados, G.: Math. Ann. Vol. 48 (1897) pp. 417–424.
Loewy, A.: S.-B. Heidelberg. Akad. Wiss. Vol. 5 (1918) p. 3.
Loewy, A.: Math. Z. Vol. 7 (1920) pp. 58–125.
Frobenius: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.
Proof by O. Perron: Math. Ann. Vol. 64 (1906) pp. 248–263.
Sylvester: C. R. Acad. Sci., Paris Vol. 98 I (1884) pp. 471–475.
Laguerre: J. École polytechn. Vol. 25 (1867) pp. 215–264.
Hensel, K.: J. reine angew. Math. Vol. 127 (1904) pp. 116–166.
Borchardt, C. W.: J. reine angew. Math. Vol. 30 (1846) pp. 38–46.
Borchardt, C. W.: J. Math, pures appl. I Vol. 12 (1847) pp. 50–67.
Sylvester, J. J.: Nouv. Ann. math. Vol. 11 (1852) pp. 439–440.
Spottiswoode, W.: J. reine angew. Math. Vol. 51 (1856) pp. 209–271 and 328-381.
Frobenius, G.: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.
Frobenius, G.: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.
Spottiswoode: C. R, Acad. Sci., Paris Vol. 94 (1882) pp. 55–59.
Frobenius, G.: Philos. Mag. V Vol. 16 (1883) pp. 267–269.
Bromwich, T.J. I’A.: Proc. Cambridge Philos. Soc. Vol. 11 (1901) pp.75 to 89.
Sylvester: Philos. Mag. V Vol. 16 (1883) pp. 267–269.
Thurston, H. S.: Amer. Math. Monthly Vol. 38 (1931) pp. 322–324.
Châtelet, A.: Ann. École norm. III Vol. 28 (1911) pp. 105–202.
Lipschitz, R.: Acta math. Vol. 10 (1887) pp. 137–144.
Ranum, A.: Bull. Amer. Math. Soc. II Vol. 17 (1911) pp. 457–461.
Taber, H.: Amer. J. Math. Vol. 13 (1891) pp. 159–172.
Bennett, A.A.: Ann. of Math. II Vol. 23 (1923) pp. 91–96.
Franklin, P.: Ann. of Math. II Vol. 23 (1923) pp. 97–100.
Franklin, P.: J. Math. Physics, Massachusetts Inst. Technol. Vol. 10 (1932) pp. 289–314.
Williamson, J.: Amer. Math. Monthly Vol. 39 (1932) pp. 280–285.
Pierce, T. A.: Bull. Amer. Math. Soc. Vol. 36 (1930) pp. 262–264.
Hermite, C.: C. R, Acad. Sci., Paris Vol. 41 (1855) PP-181–183.
Loewy, A.: J. reine angew. Math. Vol. 122 (1900) pp. 53–72.
Autonne, L.: Rend. Circ. mat. Palermo Vol. 16 (1902) pp. 104–128.
Voss, A.: Math. Ann. Vol. 13 (1878) pp. 320–374.
Prym, F.: Abh. Ges. Wiss. Göttingen Vol. 38 (1892) pp. 1–42.
Autonne: Bull. Soc. Math. France Vol. 30 (1902) pp. 121–134.
Theorem and proof by A. Hirsch: Acta math. Vol. 25 (1901) pp. 367–370.
Hermite: C. R, Acad. Sci., Paris Vol. 41 (1855) pp 181–183.
Clebsch, A.: J. reine angew. Math. Vol. 62 (1863) pp. 232–245.
This theorem and proof are by A. Hirsch: J. reine angew. Math. Vol. 62 (1863) pp. 232–245.
Bendixson, I.: Acta math. Vol. 25 (1901) pp. 359–365.
The real case by Bendixson, the complex case by Hirsch: Acta math. Vol. 25 (1901) pp. 359–365.
Bromwich: Acta math. Vol. 30 (1906) pp. 295–304.
Schur, I.: Math. Ann. Vol. 66 (1909) pp. 488–510.
Browne, E, T.: Bull. Amer. Math. Soc. Vol. 34 (1928) pp. 363–368.
Wedderburn, J. H. M.: Ann. of Math. II Vol. 27 (1926) pp. 245–248.
Aramata, H.: Tôhoku Math. J. Vol. 28 (1927) p. 281.
Brauer, R.: Tôhoku Math. J. Vol. 30 (1928) p. 72.
Rights and permissions
Copyright information
© 1933 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mac Duffee, C.C. (1933). The characteristic equation. In: The Theory of Matrices. Ergebnisse der Mathematik und Ihrer Grenƶgebiete, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-99234-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-99234-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-98421-1
Online ISBN: 978-3-642-99234-6
eBook Packages: Springer Book Archive