The quaternionic determinant

The determinant for complex matrices cannot be extended to quaternionic matrices. Instead, the Study determinant and the closely related q-determinant are widely used. We show that the Study determinant can be characterized as the unique functional which extends the absolute value of the complex determinant and discuss its spectral and linear algebraic aspects.

[1]  S. Leo,et al.  Right eigenvalue equation in quaternionic quantum mechanics , 2000, math-ph/0002051.

[2]  D. Finkelstein,et al.  NOTES ON QUATERNION QUANTUM MECHANICS. (PART II) , 1959 .

[3]  Samuel Eilenberg,et al.  The “fundamental theorem of algebra” for quaternions , 1944 .

[4]  P. Cohn.  The similarity reduction of matrices over a skew field , 1973 .

[5]  David Finkelstein,et al.  Quaternionic Representations of Compact Groups , 1963 .

[6]  S. Leo,et al.  Quantum mechanics: From complex to complexified quaternions , 1997 .

[7]  INVERSE MATRIX AND PROPERTIES OF DOUBLE DETERMINANT OVER QUATERNION FIELD , 1991 .

[8]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[9]  Samuel Schiminovich,et al.  Foundations of Quaternion Quantum Mechanics , 1962 .

[10]  Andrew Baker,et al.  Right eigenvalues for quaternionic matrices: A topological approach , 1999 .

[11]  Fuzhen Zhang.  Quaternions and matrices of quaternions , 1997 .

[12]  Quaternionic Groups in Physics: A Panoramic Review , 1999, hep-th/9905125.

[13]  R. Kippenhahn.  Über den Wertevorrat einer Matrix , 1951 .

[14]  Stefano De Leo.  Quaternions for GUTs , 1996 .

[15]  S. Leo,et al.  Translations between Quaternion and Complex Quantum Mechanics , 1994, hep-th/9401009.

[16]  Louise A. Wolf.  Similarity of matrices in which the elements are real quaternions , 1936 .

[17]  Chia-Hsiung Tze,et al.  On the Role of Division, Jordan and Related Algebras in Particle Physics , 1996 .

[18]  Ivan Niven,et al.  Equations in Quaternions , 1941 .

[19]  J. Dieudonné,et al.  Les déterminants sur un corps non commutatif , 1943 .

[20]  S. Adler,et al.  Quaternionic quantum mechanics and quantum fields , 1995 .

[21]  J. L. Brenner,et al.  Matrices of quaternions. , 1951 .

[22]  Ivan Niven.  The Roots of a Quaternion , 1942 .

[23]  Stefano De Leo,et al.  Odd Dimensional Translation between Complex and Quaternionic Quantum Mechanics , 1996 .

[24]  Helmer Aslaksen,et al.  Quaternionic determinants , 1996 .

[25]  G. Dixon,et al.  Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics , 1994 .

[26]  Stefano De Leo.  Quaternions and special relativity , 1996 .

[27]  A. S. Ramsey.  The Mathematical Papers of Sir William Rowan Hamilton , 1931, Nature.

[28]  Stefano De Leo,et al.  Quaternionic electroweak theory , 1996 .

[29]  J. Brenner.  Applications of the Dieudonné determinant , 1968 .

[30]  N. Wiegmann,et al.  Some Theorems On Matrices With Real Quaternion Elements , 1955, Canadian Journal of Mathematics.

[31]  William Rowan Hamilton,et al.  Elements of Quaternions , 1969 .

[32]  N. Johnson,et al.  Quaternionic modular groups , 1999 .