Controllability of certain real symmetric matrices with application to controllability of graphs

If M is an n x n real symmetric matrix and b is a real vector of length n, then the pair (M, b) is said to be controllable if all the eigenvalues of M are simple and M has no eigenvector orthogonal to b. Simultaneously, we say that M is controllable for b. There is an extensive literature concerning controllability of specified matrices, and in the recent past the matrices associated with graphs have received a great deal of attention. In this paper, we restate some known results and establish new ones related to the controllability of similar, commuting or Gram matrices. Then we apply the obtained results to get an analysis of controllability of some standard matrices associated with (particular) graphs.