Two classes of symmetric sign patterns that require unique inertia.
Sign pattern matrices (sign patterns) \(A\) of order \(n\), whose entries are from the set \(\{+, -, 0\}\) are studied. A class of real matrices, called the qualitative class of \(A\) and denoted by \(Q(A)\) is associated with each \(A\) from the set of all \(n \times n\) sign patterns. A sign pattern \(A \in Q(A)\) is a double star (respectively cycle) sign pattern if the undirected graph of \(A\) is a double star (respectively cycle) graph, possibly with loops. The inertia of the real symmetric matrix \(A\), written as \(i(A)\) is the set \(i(A)= \{i(B)=(i_+(B), i_-(B), i_0(B)) \mid B=B^T \in Q(A)\}\), where \(i_+(B)\) (respectively \(i_-(B)\), \(i_0(B)\)) denotes the number of positive (respectively negative, zero) eigenvalues of a matrix \(B\) counted with their algebraic multiplicities. In a special case, when \(i(B_1)=i(B_2)\) for all real symmetric matrices \(B_1, B_2 \in Q(A)\), it is said that the symmetric sign pattern \(A\) requires unique inertia. \textit{F. J. Hall}, \textit{Zhongshan Li } and \textit{Di Wang} [Linear Algebra Appl. 338, 153-169 (2001; Zbl 0994.15028)] have given sufficient and necessary conditions when a symmetric sign pattern requires unique inertia. These conditions are difficult to fulfill and in this work the special cases which characterize the double star and cycle sign patterns requiring unique inertia are presented and respective unique inertia are found.
scientific article; zbMATH DE number 2005196
scientific article