Properties of symmetric matrices (Q1771395)

The two main results in this paper describe the set of extreme and smooth points of the unit ball of the Banach space of real symmetric \( n \times n \)-matrices, where the norm of a matrix is defined as the maximum absolute value of its eigenvalues. It is proved that a symmetric real matrix is an extreme point of the unit ball if and only if all its eigenvalues are \(\pm{1}\). The smooth points of the unit ball are those elements that have exactly one eigenvalue which is \(\pm{1}\).