Symmetry Preserving Linear Sections of the Generic Symmetric Matrix

This chapter is about linear sections of the generic symmetric matrix that are symmetric themselves. While for linear sections of the \(m\times m\) generic matrix, we could use the resources of the m-generic property with all its spin-offs discussed in a previous chapter, here no such help is available. To replace some of those arguments, we resort to the use of initial ideals as based on a couple of basic related lemmas in Chapter 2 . One discusses similar families of linear sections as in the previous chapter, stressing the symmetry. Most proofs are redone ab initio due to the peculiarities of symmetry in those linear sections. The discussion of the main algebraic invariants becomes an interesting challenge vis-à-vis the behavior of its analogs in the case of linear sections of the generic matrix. By and large, a good acquaintance with the previous chapter may help advancing through the present chapter, helping to get a grasp of the main similarities and differences in the theory. As a natural fallout, this chapter is shorter than the previous one.