4. phasors and eigenvalues suppose that we have the two-dimensional system of differential equations expressed in matrix/vector form: d dt ~x(t) = a~x(t) +~bu(t) (1) where for this problem, the matrix a and the vector~b are both real. (a) give a necessary condition on the eigenvalues λi of a such that any impact of an initial condition will eventually completely die out. (i. e. the system will reach steady-state.) you don’t have to prove this. (we will see a detailed proof later in the course.) the idea here is to make sure that you understand what kind of thing is required. (b) now assume that u(t) has a phasor representation ue. in other words, u(t) = ue e +jωt +ue e −jωt . assume that the vector solution ~x(t) to the system of differential equations (1) can also be written in phasor form as ~x(t) =~ xee +jωt + ~ xee −jωt . (2) derive an expression for ~ xe involving a,~b, jω, ue, and the identity matrix i. (hint: plug (2) into (1) and simplify, using the rules of differentiation and grouping terms by which exponential e±jωt they multiply. )