Neutral subspaces of pairs of symmetric/skewsymmetric real matrices

Let A and B be n × n real matrices with A symmetric and B skewsymmetric. Obviously, every simultaneously neutral subspace for the pair (A,B) is neutral for each Hermitian matrix X of the form X = μA + iλB, where μ and λ are arbitrary real numbers. It is well-known that the dimension of each neutral subspace of X is at most In+(X) + In0(X), and similarly, the dimension of each neutral subspace of X is at most In−(X) + In0(X). These simple observations yield that the maximal possible dimension of an (A,B)-neutral subspace is no larger than min{min{In+(μA+ iλB) + In0(μA+ iλB), In−(μA+ iλB) + In0(μA+ iλB)}}, where the outer minimum is taken over all pairs of real numbers (λ, μ). In this paper, it is proven that the maximal possible dimension of an (A,B)-neutral subspace actually coincides with the above expression.