×

zbMATH — the first resource for mathematics

Quaternions and matrices of quaternions. (English) Zbl 0873.15008
The author gives a useful survey on quaternions and matrices of quaternions. He recalls standard facts going back to Rowan Hamilton as well as new results motivated by applications in physical theories. The main research problem presented in the paper is to extend the classical matrix theory from complex to the quaternion matrices.

MSC:
15B33 Matrices over special rings (quaternions, finite fields, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adler, S.L., Quaternionic quantum mechanics and quantum fields, (1994), Oxford U.P New York
[2] Apostol, T.M., Mathematical analysis, (1977), Addison-Wesley · Zbl 0126.28202
[3] Y. H. Au-Yeung, A short proof of a theorem on the range of a normal quaternionic matrix, Linear and Multilinear Algebra, to appear. · Zbl 0838.15017
[4] Y. H. Au-Yeung, 1994. On the eigenvalues and numerical range of a quaternionic matrix, Preprint. · Zbl 0903.15004
[5] Au-Yeung, Y.H., On the convexity of numerical range in quaternionic Hilbert spaces, Linear and multilinear algebra, 16, 93-100, (1984) · Zbl 0563.15016
[6] Brown, W.C., Matrices over commutative rings, (1992), Marcell Dekker
[7] Birkhoff, G.; MacLane, S., A survey of modern algebra, (1977), Macmillan · Zbl 0365.00006
[8] Brenner, J.L., Matrices of quaternions, Pacific J. math., 1, 329-335, (1951) · Zbl 0043.01402
[9] Bunse-Gerstner, A.; Byers, R.; Mehrmann, V., A quaternion QR algorithm, Numer. math., 55, 83-95, (1989) · Zbl 0681.65024
[10] Cao, C.G., The numerical radius of real quaternion matrix, J. xinjiang univ. 7, No. 2, (1990), also see MR 89k:15031
[11] Chen, L.X., Definition of determinant and cramer solutions over the quaternion field, Acta math. sinica N. S., 7, 2, 171-180, (1991) · Zbl 0742.15002
[12] Chen, L.X., The extension of Cayley-Hamilton theorem over the quaternion field, Chinese sci. bull., 17, 1291-1293, (1991)
[13] Chen, L.X., Inverse matrix and properties of double determinant over quaternion field, Sci. China ser. A, 34, 5, 528-540, (1991) · Zbl 0741.15005
[14] Cohn, P.M., Skew field constructions, () · Zbl 0355.16009
[15] Cohn, P.M., The similarity reduction of matrices over a skew field, Math. Z., 132, 151-163, (1973) · Zbl 0248.15010
[16] Eilenberg, S.; Niven, I., The “fundamental theorem of algebra” for quaternions, Bull. amer. math. soc., 50, 246-248, (1944) · Zbl 0063.01228
[17] Eilenberg, S.; Steenrod, N., Foundations of algebraic topology, (1952), Princeton U.P · Zbl 0047.41402
[18] Finkelstein, D.; Jauch, J.M.; Schiminovich, S.; Speiser, D., Foundations of quaternion quantum mechanics, J. math. phys., 3207-3220, (1962)
[19] Finkelstein, D.; Jauch, J.M.; Speiser, D., Notes on quaternion quantum mechanics, Cern, 59, 9, 59, (1959)
[20] Guerlebeck, K.; Sproessig, W., Quaternionic analysis, (1989), Akademie Verlag Berlin
[21] Halmos, P.R., A Hilbert space problem book, (1967), Van Nostrand · Zbl 0144.38704
[22] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge U.P · Zbl 0576.15001
[23] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1994), Cambridge U.P · Zbl 0801.15001
[24] Jacobson, N., Basic algebra I, (1974), W. H. Freeman · Zbl 0284.16001
[25] Jamison, J.E., Numerical range and numerical radius in quaternionic Hilbert spaces, ()
[26] Johnson, R.E., On the equation χα = γχ + β over an algebraic division ring, Bull. amer. math. soc., 50, 202-207, (1944) · Zbl 0061.05505
[27] Von Kippenhahn, R., Über der wertevorrat einer matrix, Math. nachr., 6, 3/4, 193-228, (1951) · Zbl 0044.16201
[28] Krieg, A., Modular forms on half-spaces of quaternions, () · Zbl 0564.10032
[29] Lee, H.C., Eigenvalues of canonical forms of matrices with quaternion coefficients, (), Sect. A · Zbl 0036.29802
[30] Mehta, M.L., Determinants of quaternion matrices, J. math. phys. sci. 8, No. 6, (1974) · Zbl 0319.15005
[31] N. Mackey, Hamilton and Jacobi meet again: Quaternions and the eigenvalue problem, SIAM J. Matrix Anal. Appl., to appear. · Zbl 1280.65035
[32] Niven, I., Equations in quaternions, Amer. math. monthly, 48, 654-661, (1995) · Zbl 0060.08002
[33] Steenrod, N., The topology of fibre bundles, (1951), Princeton, U.P · Zbl 0054.07103
[34] So, W., Left eigenvalues of quaternionic matrices, (1994), a Private communication
[35] So, W.; Thompson, R.C., Convexity of the upper complex plane part of the numerical range of a quaternionic matrix, (1995), Manuscript
[36] So, W.; Thompson, R.C.; Zhang, F., Numerical ranges of matrices with quaternion entries, Linear and multilinear algebra, 37, 175-195, (1994) · Zbl 0814.15026
[37] Viswanath, K., Normal operators on quaternionic Hilbert spaces, Trans. amer. math. soc. 162, (1971) · Zbl 0234.47024
[38] Viswanath, K., Contributions to linear quaternionic Hilbert analysis, () · Zbl 0234.47024
[39] Wolf, L.A., Similarity of matrices in which the elements are real quaternions, Bull. amer. math. soc., 42, 737-743, (1936) · JFM 62.1078.03
[40] Wiegmann, N.A., Some theorems on matrices with real quaternion elements, Canad. J. math., 7, 191-201, (1955) · Zbl 0064.01604
[41] Wood, R.M.W., Quaternionic eigenvalues, Bull. London math. soc., 17, 137-138, (1985) · Zbl 0537.15011
[42] Xie, B.J., An extension of Hadamard’s theorem over the skew field of quaternions, Sci. sinica, 88-93, (1970), also see MR 83g:15014
[43] Xie, B.J., Applications of characteristic roots and standard forms of matrices over a skew field, Acta math. sinica, 23, 4, 522-533, (1980), also see MR 82m:15021 · Zbl 0443.15008
[44] Xie, B.J., Existence theorems for the canonical diagonal form and the weak canonical form of characteristic matrices over a skew field, Acta math. sinica, 23, 3, 398-410, (1980), also see MR 82k:15015 · Zbl 0436.15010
[45] Xie, B.J., An expansion theorem for determinants of self-adjoint quaternion matrices and its applications, Acta math. sinica, 23, 5, 668-683, (1980), also see MR 83a:15007 · Zbl 0466.15007
[46] Xie, B.J., Several theorems concerning self-conjugate matrices and determinants of quaternions, Acta sci. natur. univ. jilin, No. 1, 1-7, (1982), also see MR 84c:15022
[47] Zhang, F., Permanent inequalities and quaternion matrices, ()
[48] Zhang, F., On numerical range of normal matrices of quaternions, J. math. physical sciences, 29, 6, 235-251, (1995) · Zbl 0855.15006
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.