A small note on the scaling of symmetric positive definite semiseparable matrices

Abstract

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix $$T$$ satisfies property A, one can easily construct a diagonal matrix $$\hat{D}$$ such that $$\hat{D}T\hat{D}$$ has the lowest condition number over all matrices $$DTD$$ , for any choice of diagonal matrix $$D$$ . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form.