# Properties of matrices with numerical ranges in a sector

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Shanghai University‎, ‎Shanghai 200444‎, ‎China.

Abstract

Let $(A)$ be a complex $(n\times n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(\alpha)$ denoted by $(S_{\alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{\alpha})$.
The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruent matrix and unitary matrix of polar decompostion are also included in the same sector. Furthermore, we extend some inequalities about eigenvalues and singular values and the linear fractional maps to this class of matrices.

Keywords

Main Subjects

### References

Yu.M. Arlinskii and A.B. Popov, On sectorial matrices, Linear Algebra Appl. 370 (2003) 133--146.

R. Bhatia and X.Z. Zhan, Compact operator whose real and imaginary parts are positive, Proc. Amer. Math. Soc. 129 (2001) 2277--2281.

S. Drury, Principal powers of matrices with positive definite real part, Linear Multilinear Algebra 63 (2015) 296--301.

S. Drury and M. Lin, Singular value inequalities for matrices with numerical ranges in a sector, Oper. Matrices 8 (2014) 1143--1148.

S.W. Drury, Fischer determinantal inequalities and Higham's Conjecture, Linear Algebra Appl. 439 (2013) 3129--3133.

X.H. Fu and Y. Liu, Rotfel'd inequality for partitioned matrices with numerical ranges in a sector, Linear Multilinear Algebra 64 (2016) 105--109.

A. George and Kh. D. IKramov, On the properties of Accretive-Dissipative Matrices, Math. Notes 77 (2005) 767--776.

A. George, Kh.D. Ikramov and A.B. Kucherov, On the growth factor in Gaussian elimination for generalized Higham matrices, Numer. Linear Algebra Appl. 9 (2002) 107--114.

N.J. Higham, Factorizing complex sysmmetric matrices with positive real and imaginary parts, Math. Comp. 67 (1998) 1591--1599.

R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1994.

R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, 1985; Russian translation; Mir, Moscow, 1990.

Kh.D. Ikramov, Determinantal Inequalities for Accretive-Dissipative Matrices, J. Math. Sci. 121 (2004) 2458--2464

Kh.D. Ikramov and V.N. Chugunov, Inequalities of Fisher and Hadamard types for accretive-dissipative matrices, Dokl. Ross. Akad. Nauk 384 (2002) 585--586

Kh.D. Ikramov and A.B. Kucherov, Bounding the growth factor in Gaussian elimination for Buchley's class of complex symmetric matrices, Numer. Linear Algebra Appl. 7 (2000) 269--274.

C.K. Li and N.S. Sze, Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector, J. Math. Anal. Appl. 410 (2014) 487--491.

M. Lin, Reversed determinantal inequalities for accretive-dissipative matrices, Math. Inequal. Appl. 15 (2012) 955--958.

M. Lin, Fischer type determinantal inequalities for accretive-dissipative matrices, Linear Algebra Appl. 438 (2013) 2808--2812.

M. Lin, Extension of a result of Haynsworth and Hartel, Arch. Math. 104 (2015) 93--100.

M. Lin, Some inequalities for sector matrices, Oper. Matrices 10 (2016), no. 4, 915--921.

R. Mathias, Matrices with positive-definite Hermitian part: Inequalities and linear systems, SIAM J. Matrix Anal. Appl. 13 (1992) 640--654.

F. Zhang, A matrix decomposition and its application, Linear Multilinear Algebra 63 (2015) 2033--2042.

P.P. Zhang, Extension of Matic's results, Linear Algebra Appl. 486 (2015) 328--334.

### History

• Receive Date: 13 May 2016
• Revise Date: 21 September 2016
• Accept Date: 21 September 2016