Simultaneous symmetrization (Q1826838)

Let us define by \(V\) the real linear subspace of \(n\times n\) matrices such that each matrix from \(V\) is similar to a real symmetric matrix. The author proves that the maximal dimension of the linear space \(V\) is \(n\left( n+1\right) /2\), and if the dimension of the subspace is maximal then there exist an invertible real matrix \(P\) such that \(P^{-1}VP=\left\{ P^{-1}AP;A\in V\right\} =\Im _{n}\) i.e. the space \(V\) is simultaneously symmetrizable. As open problem still remains the simulataneous symmetrization of a linear space of real \ matrices, such that each of them is similar to a real symmetric matrix when the dimension of the space is less than \(n\left( n+1\right) /2\).