Principal submatrices .VII. Further results concerning matrices with equal principal minors

Abstract

~This paper c haracte rizes real s ymmetric matri ces A s uc h that all t X t principal minors a re equal and a ll t X t nonprincipal minors a re of fix ed sign , for two consecutive values of t less than rank A. It a lso c ha racte riz es matrices A (over a n arbitrary fi eld ) in which a ll t X t principal minors are e qua l and all nonprin cipal t X t minors a re e qual, for o ne fix ed value of t less than rank A. In the paper Principal Submatrices V, [5],1 a classifi cati o n was found for symme tri c matrices A for whic h all t X t principal minors of A are e qual, for three consecutive values of t less than the rank of A. It is the purpose of thi s pape r to present a similar theorem classifying the real symme tri c matrices in whic h the condition on the principal minors is weake ned to requirin g that all t X t prin­ cipal minors of A be e qual , for two consecuitive values of t less than the rank of A, and in whic h a s ign condition is imposed on the . nonprin cipal tX t minors for these two consecutive values of t . This result is presented in Theore m 1. In thi s paper we also classify all square matrices A (over an arbitra ry field and not necessarily symme tri c) in whic h the condition on the principal minors of A is weakened to re quirin g that all t X t princi pal minors of A be equal for one value of t less than the rank of A , and for thi s value of t the condition on the no n principal t X t minors of A is s tre ngth­ e ned to requirin g that they all be equal. This r e,s ult is presented in Theore m 4. THEOREM 1. Let r be a fixed integer and let A be an n X n symmetric matrix over the real number field, such that: