(Preservation of Eigenvalues under Similarity Transform.). Consider a matrix AЄ Rx, and a non-singular matrix T = Rx. Show that the eigenvalues of B = TAT-1 are the same as those of A. Remark. This important fact in linear algebra is the basis for the similarity transform that a redefinition of the state (to a new set of state variables in which the equations above may have simpler representation) does not affect the eigenvalues of the A matrix, and thus the stability of the system. We will use this similarity transform in our analysis of linear systems.