Determinantal Inequalities for TN Matrices

This chapter begins an investigation into the possible relationships among the principal minors of TN matrices. An important goal is to examine all the inequalities among the principal minors of TN matrices. More generally, this chapter also studies families of determinantal inequalities that hold for all TN matrices. The intention here is to employ certain ideas pertinent to theory of TN matrices to derive a description with respect to positive semidefinite matrices. One notable difference between the class of TN matrices and the positive semidefinite matrices is that TN matrices are not in general closed under the simultaneous permutation of rows and columns, while positive semidefinite matrices are closed under such an operation.