References
E. Artin,Geometric Algebra, New York: Interscience, 1957; reprinted by Wiley, New York, 1988.
H. Aslaksen,SO(2) invariants of a set of 2 x 2 matrices.Math. Scand. 65 (1989), 59–66.
H. Aslaksen, E.-C. Tan, and C. Zhu, Invariant theory of special orthogonal groups,Pacific ]. Math. (in press).
A. Bagazgoitia, A determinantal identity for quaternions, inProceedings of 1983 Conference on Algebra Lineal y Aplicaciones, Vitoria-Gasteiz, Spain, 1984, pp. 127–132.
R. W. Barnard and E. Hastings Moore,General analysis. Part 1, Memoirs of the American Philosophical Society, 1935.
J. Brenner, Expanded matrices from matrices with complex elements,SIAM Rev. 3 (1961), 165–166.
J. Brenner, Applications of the Dieudonné determinant,Linear Algebra Appl. 1 (1968), 511–536.
J. Brenner, Corrections to “Applications of the Dieudonné determinant,”Linear Algebra Appl. 13 (1976), 289.
J. Brenner and J. De Pillis, Generalized elementary symmetric functions and quaternion matrices,Linear Algebra Appl. 4 (1971), 55–69.
A. Cayley, On certain results relating to quaternions,Philos. Mag. 26 (1845), 141–145; reprinted inThe Collected Mathematical Papers Vol. 1, Cambridge: Cambridge University Press, 1989, pp. 123–126.
L. Chen, Definition of determinant and Cramer solution over the quaternion field,Acta Math. Sinica (N.S.) 7 (1991), 171–180.
L. Chen, Inverse matrix and properties of double determinant over quaternion field,Sci. China Ser. A 34 (1991), 528–540.
P. M. Cohn, The similarity reduction of matrices over a skew field,Math. Z. 132 (1973), 151–163.
P. M. Cohn,Algebra, vol. 1, 2nd ed., New York: Wiley, 1991.
P. M. Cohn,Algebra, vol. 3, 2nd ed. New York: Wiley, 1991.
M. L. Curtis,Matrix Groups, New York: Springer-Verlag, 1979; 1984.
J. Dieudonné, Les déterminants sur un corps non-commutatif,Bull. Soc. Math. France 71 (1943), 27–45.
J. Dieudonné.Special Functions and Linear Representations of Lie Groups, CBMS 42, Providence, RI, American Mathematical Society, 1980.
R. Dimitrić and B. Goldsmith, Sir William Rowan Hamilton,Math. Intelligencer 11 (1989), no. 2, 29–30.
F. J. Dyson, Correlations between eigenvalues of a random matrix,Commun. Math. Phys. 19 (1970), 235–250.
F. J. Dyson, Quaternion determinants,Helv. Phys. Acta 45 (1972), 289–302.
I. M. Gelfand and V. S. Retakh, Determinants of matrices over noncommutative rings,Functional Anal. Appl. 25 (1991), 91–102.
F. Reese Harvey,Spinors and Calibrations, New York, Academic Press, 1990.
W. R. Hamilton,Elements of Quaternions, 2nd ed., London: Longman, 1889.
A. Heyting, Die Théorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplikation,Math. Ann. 98 (1927), 465–490.
M. H. Ingraham, A note on determinants,Bull. Amer. Math. Soc. 43 (1937), 579–580.
N. Jacobson, Normal semi-linear transformations,Amer. J. Math. 61 (1939), 45–58.
N. Jacobson, An application of E. H. Moore’s determinant of a Hermitian matrix,Bull. Amer. Math. Soc. 45 (1939), 745–748.
H. C. Lee, Eigenvalues and canonical forms of matrices with quaternionic entries,Proc. Roy. Irish Acad. Sect. A, 52 (1949), 253–260.
D. W. Lewis, A determinantal identity for skewfields,Linear Algebra Appl. 7 (1985), 213–217.
K. O. May, The impossibility of a division algebra of vectors in three dimensional space,Amer. Math. Monthly 73 (1966), 289–291.
Madan Lal Mehta, Determinants of quaternion matrices,J. Math. Phys. Sci. 8 (1974), 559–570.
Madan Lal Mehta,Elements of Matrix Theory, Dehli Hindustan Pub. Corp., 1977.
E. H. Moore, On the determinant of an hermitian matrix of quaternionic elements,Bull. Amer. Math. Soc. 28 (1922), 161–162.
Thomas Muir,The Theory of Determinants, Vol. 2, London: MacMillan, 1911.
O. Ore, Linear equations in non-commutative fields,Ann. Math. 32 (1931), 463–77.
K. Hunger Parshall and D. E. Rowe,The Emergence of the American Mathematical Research Community, 1876-1900: J. J. Sylvester, Felix Klein and E. H. Moore, Providence, RI: American Mathematical Society, 1994.
J. M. Peirce, Determinants of quaternions,Bull. Amer. Math. Soc. 5 (1899), 335–337.
P. Piccinni, Dieudonné determinant and invariant real polynomials on gl(n, H),Rend. Mat. (7)2 (1982), 31–45.
R. S. Pierce,Associative Algebras, New York: Springer-Verlag, 1982.
J. Radon, Lineare Scharen orthogonaler Matrizen,Abh. Math. Sem. Univ. Hamburg 1 (1922), 2–14.
A. R. Richardson, Hypercomplex determinants,Messenger of Math. 55 (1926), 145–152.
A. R. Richardson, Simultaneous linear equations over a division algebra,Proc. London Math. Soc. 28 (1928), 395–420.
E. Study, Zur Théorie der linearen Gleichungen,Acta Math. 42 (1920), 1–61.
O. Teichmüller, Operatoren im Wachsschen Raum,J. Reine Angew. Math. 174 (1935), 73–124.
C. L. Tong, Symplectic Groups, honours thesis, National Univ. of Singapore, 1991.
B. Leednert van der Waerden, Hamilton’s discovery of quaternions,Math. Mag. 49 (1976), 227–234.
B. Leednert van der Waerden,A History of Algebra, New York: Springer-Verlag, 1985.
P. Van Praag, Sur les déterminants des matrices quaterniennes,Helv. Phys. Acta 62 (1989), 42–46.
P. Van Praag, Sur la norme réduite du déterminant de Dieudonné des matrices quaterniennes,J. Algebra 136 (1991), 265–274.
L. A. Wolf, Similarity of matrices in which the elements are real quaternions,Bull. Amer. Math. Soc. 42 (1936), 737–743.
L. E. Zagorin, The determinants of matrices over a field (Russian),Proc. First Republican Conf. Math. Byelorussia, Izdat, Minsk: “Vysšaja Škola”, 1965, pp. 151–152.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aslaksen, H. Quaternionic determinants. The Mathematical Intelligencer 18, 57–65 (1996). https://doi.org/10.1007/BF03024312
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03024312