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We study certain pairs of subspaces V and W of C n we call supports that consist of eigenspaces of the eigenvalues ± ‖ M ‖ of a minimal hermitian matrix M (‖ M ‖ ≤ ‖ M + D ‖ for all real diagonals D). For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds. [ABSTRACT FROM AUTHOR]

Abstract: In general -algebras, elements with minimal norm in some equivalence class are introduced and characterized. We study the set of minimal hermitian matrices, in the case where the -algebra consists of complex matrices, and the quotient is taken by the subalgebra of diagonal matrices. We thoroughly study the set of minimal matrices particularly because of its relation to the geometric problem of finding minimal curves in flag manifolds. For the flag manifold of ‘four mutually orthogonal complex lines’ in , it is shown that there are infinitely many minimal curves joining arbitrarily close points. In the case of the flag manifold of ‘three mutually orthogonal complex lines’ in , we show that the phenomenon of multiple minimal curves joining arbitrarily close points does not occur. [Copyright &y& Elsevier]