zbMATH — the first resource for mathematics

Cramer’s rule for quaternionic systems of linear equations. (English. Russian original) Zbl 1157.15308
J. Math. Sci., New York 155, No. 6, 839-858 (2008); translation from Fundam. Prikl. Mat. 13, No. 4, 67-94 (2007).
Summary: New definitions of determinant functionals over the quaternion skew field are given. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer’s rules for right and left quaternionic systems of linear equations are obtained.

15A15 Determinants, permanents, traces, other special matrix functions
15A06 Linear equations (linear algebraic aspects)
15B33 Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
[1] H. Aslaksen, ”Quaternionic determinants,” Math. Intelligencer, 18, No. 3, 57–65 (1996). · Zbl 0881.15007 · doi:10.1007/BF03024312
[2] L. Chen, ”Definition of determinant and Cramer solutions over quaternion field,” Acta Math. Sinica (N.S.), 7, No. 2, 171–180 (1991). · Zbl 0742.15002
[3] L. Chen, ”Inverse matrix and properties of double determinant over quaternion field,” Sci. China Ser. A, 34, 528–540 (1991). · Zbl 0741.15005
[4] N. Cohen and S. De Leo, ”The quaternionic determinant,” Electron. J. Linear Algebra, 7, 100–111 (2000). · Zbl 0977.15004
[5] F. J. Dyson, ”Quaternion determinants,” Helv. Phys. Acta, 45, 289–302 (1972).
[6] I. Gelfand and V. Retakh, ”A determinants of matrices over noncommutative rings,” Funkts. Anal. Prilozh., 25, No. 2, 13–35 (1991).
[7] I. Gelfand and V. Retakh, ”A theory of noncommutative determinants and characteristic functions of graphs,” Funkts. Anal. Prilozh., 26, No. 4, 33–45 (1992). · Zbl 0786.06005 · doi:10.1007/BF01077071
[8] I. I. Kirchej, ”Fractional-rational regularization of a system of linear equations over the skew-field of quaternions,” J. Math. Sci., 90, No. 5, 2398–2403 (1998). · doi:10.1007/BF02433974
[9] I. I. Kyrchei, ”Classical adjoint for Hermitian matrix over quasi-field,” Mat. Metody i Fiz.-Mekh. Polya, 44, No. 3, 33–48 (2001). · Zbl 1098.15501
[10] I. I. Kyrchei, ”Analogue of adjoint matrix over skew field with involution,” Mat. Metody i Fiz.-Mekh. Polya, 46, No. 4, 81–91 (2003). · Zbl 1086.15507
[11] I. S. Ponizovsky, ”On a determinant of matrices with elements from some ring,” Mat. Sb., 45(87), No. 1, 3–16 (1958).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.