Factorizations of a matrix into symmetric matrices with prescribed nullities (Q1915607)

Let \(A\) be a square matrix over a field \(K\). Suppose \(A= S_1 S_2 \dots S_k\), \(k\geq 2\), where \(S_i\) are symmetric matrices. Let \(n\) and \(n_i\) be the nullities of \(A\) and \(S_i\), respectively. Clearly \(n\geq n_i\geq 0\) and \(n_1+ n_2+ \dots+n_k\geq n\). The author shows that for \(k\geq 3\) the converse holds. For \(k=2\) a similar result holds. For a complex square matrix \(A\), the author gives necessary and sufficient conditions for \(A\) to be a product of two Hermitian matrices with prescribed nullities.