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The determinant of a complex matrix and Gershgorin circles. (English) Zbl 1430.15006

The authors prove that for a complex matrix the set product of Gershgorin circles contains the determinant. This statement has been previously shown for real matrices.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A42 Inequalities involving eigenvalues and eigenvectors
51A20 Configuration theorems in linear incidence geometry
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References:

[1] R.T. Farouki, H.P. Moon, and B. Ravani. Minkowski geometric algebra of complex sets. Geometriae Dedicata, 85:283- 315, 2001. · Zbl 0987.51012
[2] R.T. Farouki and H. Pottmann. Exact Minkowski products of N complex disks. Reliable Computing, 8:43-66, 2002. · Zbl 1028.65048
[3] S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215-219, 2019. · Zbl 1405.15011
[4] H. Schneider. An inequality for latent roots of a matrix applied to determinants with dominant main diagonal. Journal of the London Mathematical Society, 28:8-20, 1953. · Zbl 0050.01103
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