Determinants of sums of two real matrices and their extensions

Abstract

We study the determinants of the sum of two real matrices under the action of SO(n) ⊗ SO(n). The extremal determinants are determined. The result is a refinement of the results of Li and Mathias [C.K. Li and R. Mathias, The determinant of the sum of two matrices, Bull. Aust. Math. Soc. 52 (1995), pp. 425–429]. We also study the problem in the context of real classical simple Lie algebras. The results of Fiedler [M. Fiedler, Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc. 30 (1971), pp. 27–31], Li and Mathias 4 and Tam and Thompson [T.Y. Tam and M.C. Thompson, Determinant and Pfaffian of sum of skew symmetric matrices, Linear Algebra Appl. 433 (2010), pp. 412–423] are some special cases in this context. Complete solutions are obtained.

Keywords:

• determinant
• orbit
• real classical simple Lie algebras
• adjoint orbit

AMS Subject Classifications::

• Primary: 15A15
• Secondary: 17C20

Acknowledgements

The authors are grateful to two anonymous referees for many helpful suggestions.