Convex Sets of Hermitian Matrices with Constant Inertia
Charles R. Johnson, Leiba Rodman · SIAM Journal on Algebraic and Discrete Methods · 1985
For n-by-n Hermitian matrices $A_1 , \cdots ,A_m $, the situation in which all matrices in the convex hull of $A_1 , \cdots ,A_m $, have the same inertia is studied. It is shown, for example, that if $m = 2$or$n = 2$ and the matrices are nonsingular, then they are simultaneously congruent to matrices of a special form in which the upper left principal submatrix is positive definite and its complementary principal submatrix is negative definite. The singular case is also studied, and the nonsingular case for $m > 2$, $n > 2$ remains open.