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An Explicit Criterion for the Convexity of Quaternionic Numerical Range

Published online by Cambridge University Press:  20 November 2018

Wasin So
Wasin So*
Affiliation:
Department of Mathematical and Information Sciences Sam Houston State University Huntsville, Texas 77341 U.S.A., e-mail: mth [email protected]
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Abstract

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Quaternionic numerical range is not always a convex set. In this note, an explicit criterion is given for the convexity of quaternionic numerical range.

Keywords

15A3315A60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 1 , 01 March 1998 , pp. 105 - 108
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Au-Yeung, Y. H., Another proof of the theorems on the eigenvalues of a square quaternion matrix. Proc. Glasgow Math. Assoc. 6 (1964), 191–195.Google Scholar
2. Au-Yeung, Y. H., On the convexity of numerical range in quaternionic Hilbert spaces. Linear and Multilinear Algebra 16 (1984), 93–100.Google Scholar
3. Au-Yeung, Y. H., On the eigenvalues and numerical range of a quaternionic matrix. In: Five Decades As a Mathematician and Educator: On the 80th Birthday of Professor Yung-Chow Wong, World Scientific Publishing, (1995), 19–30.Google Scholar
4. Au-Yeung, Y. H., A short proof of a theorem on the numerical range of a normal quaternionic matrix. Linear and Multilinear Algebra 39 (1995), 279–284.Google Scholar
5. Brenner, J., Matrices of quaternions. Pacific J. Math. 1 (1951), 329–335.Google Scholar
6. Eilenberg, S. and Niven, I., The fundamental theorem of algebra for quaternions. Bull. Amer.Math. Soc. 50 (1944), 246–248.Google Scholar
7. Jamison, J. E., Numerical range and numerical radius in quaternionic Hilbert space. University of Missouri Ph.D. Dissertation, 1972.Google Scholar
8. Kippenhahn, R., Uber die Wertvorrat einer Matrix. Math. Nachr. 6 (1951), 193–228.Google Scholar
9. Lee, H. C., Eigenvalues and canonical forms of matrices with quaternion coefficients. Proc. Roy. Irish Acad. Sect. A 52 (1949), 253–260.Google Scholar
10. Mackey, N., Hamilton and Jacobi meet again; quaternions and the eigenvalues problem. SIAM J. Matrix Anal. Appl. 16 (1995), 421–435.Google Scholar
11. Niven, I., Equations in quaternions. Amer. Math. Monthly 48 (1941), 654–661.Google Scholar
12. So, W. and Thompson, R. C., Convexity of the upper complex plane part of the numerical range of a quaternionic matrix. Linear and Multilinear Algebra 41 (1996), 303–365.Google Scholar
13. So, W., Thompson, R. C. and Zhang, F., The numerical range of normal matrices with quaternion entries. Linear and Multilinear Algebra 37 (1994), 175–195.Google Scholar
14. Thompson, R. C., The upper numerical range of a quaternionic matrix is not a complex numerical range. Linear Algebra Appl. 254 (1997), 19–28.Google Scholar
15. Wood, R. M., Quaternionic eigenvalues. Bull. London Math. Soc. 17 (1985), 137–138.Google Scholar
16. Zhang, F., Quaternions and matrices of quaternions. Linear Algebra Appl. 251 (1997), 21–57.Google Scholar
17. Zhang, F., On numerical range of normal matrices of quaternions. J.Math. Phys. Sci. 29(1995), 235–251.Google Scholar