COMMENTS ON QUIZ #12

1. The condition for 0 to be an asymptotically stable state is that ALL eigenvalues be strictly less than 1 in absolute value. (One does not talk about stability one eigenvalue at a time -- each part of the question has just one answer, not an answer for each eigenvalue.)

Therefore: 


(a) .9 and .7     0 is an asymptotically stable state
(b) 1.1 and .4    0 is NOT an asymptotically stable state
(c) .5 +/- .4i    0 is an asymptotically stable state, since the
                  eigenvalues have absolute value (.25 + .16)^{1/2}
                  which is < 1.
2. By definition, the matrix is 

2  4
3 -5
(Giving the transpose of this instead cost 2 points.)

3. The matrix with respect to the basis of columns of S is S^{-1} A S which is the product 


                  
-2  1  4    4 1 9     1 2 3
 3 -2 -5    7 2 5    -1 0 2
-1  1  2    3 6 0     1 1 1
(Reversing the roles of S^{-1} and S cost 2 points.)