A consideration on the condition number of extremely ill-conditioned matrices

As for a matrix A we examine two problems: (a) To find the upper and the lower bound of Cond<sub>2</sub>(A) in terms of two coefficients p<sub>1</sub> and p<sub>n-1</sub> (see Section 2) of the characteristic polynomial of AA<sup>T</sup>, and (b) proof of existence of a matrix A having considerably larger condition number than that obtained in the previous papers. The connection between (a) and (b) is illustrated.