Consider the first order differential equation (1) , where is an complex matrix. A solution of (1) (indeed, of any differential equation) is called a steady state if it is constant in . A steady state is called globally asymptotically stable if for any other solution , we have . (Limits are taken entrywise.) See pages 507-508 of D&F for a more lucid explanation of steady states and globally asymptotically stable steady states.
Prove that if the eigenvalues of have negative real parts, then the zero solution of is globally asymptotically stable.
[My usual disclaimer about analytical problems applies. Be on the lookout for Blatantly Silly Statements.]
As we have seen, every solution of (1) is a linear combination of the columns of . If is in Jordan canonical form, then every solution of (1) is a linear combination of the columns of . Note the exponential of a Jordan block as we computed here; in particular, we can see that if is a solution of (1), then each entry of is a linear combination of functions of the form , where is an eigenvalue of and is a polynomial in . Say ; then .
Since is negative, as approaches infinity, tends to 0 faster than any polynomial, so that our solution tends to the constant solution 0.
Thus the zero solution is globally asymptotically stable.