On best conditioned matrices

1. Main theorems. Let A be a positive definite Hermitian matrix of finite order, and let A and X be its maximal and minimal eigenvalue respectively. The condition number of A is the ratio P(A) =A/X introduced by Todd [1]. Let 'G be a class of regular linear transformations. Define ATT*A T. We say that A is best conditioned with respect to 'G if P(A T) > P(A) for all TEE T. In order to investigate whether A is best conditioned we remember that