## 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.)