  A large number matrix optimization problems are described by orthogonally
invariant norms. This paper is devoted to the study of variational analysis of
the orthogonally invariant norm cone of symmetric matrices. For a general
orthogonally invariant norm cone of symmetric matrices, formulas for the
tangent cone, normal cone and second-order tangent set are established. The
differentiability properties of the projection operator onto the orthogonally
invariant norm cone are developed, including formulas for the directional
derivative and the B-subdifferential. Importantly, the directional derivative
is characterized by the second-order derivative of the corresponding symmetric
function, which is convenient for computation. All these results are specified
to the Schatten $p$-norm cone, especially to the second-order cone of symmetric
matrices.
