Minimal paramount matrices

Abstract

A real symmetric matrix of order n, n ⩾ 2, is said to be paramount if each proper principal minor is not less than the absolute value of any other minor built from the same rows. A paramount matrix is minimal ∗ ∗ In the literature the term “irreducible” has been used to describe a minimal paramount matrix. We thank the referee for his suggestion to adopt “minimal” in order to avoid confusion with other uses of “irreducible” in matrix theory. if reducing any of the diagonal entries removes the matrix from the paramount class. Minimal paramount matrices arise in the n-port realization problem of circuit theory. A condition is found that is equivalent to the minimality of a paramount matrix. Conditions are also found that guarantee that the inverse of an invertible minimal paramount matrix is itself minimal.