Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
WorkSheet 8
Ã
!
18 ¡30
1) Let A =
:
10 ¡17
(a) Find the eigenvalues (by hand) of A:
(b) Find a basis for the eigenspace of one of the eigenvalues (by hand).
2) Find the eigensystem of
Ã
2 3
0 2
!
:
Ã
!
(Ã
!)
(Ã
!)
3
4
¡25 36
2
3
3) Let A =
. Assume the eigensystem is
$ ¡1;
$ 2. Now
¡18 26
1
1
de…ne the matrix D to be the diagonal matrix whose diagonal elements are the eigenvalues
of A and de…ne S to be the matrix whose columns are the eigenvectors of A (same order as
for D and also use nicer eigenvectors than those given above).
(a) Calculate SDS ¡1 :
(b) Calculate A5 in a tricky way (ie. replace A by SDS ¡1 )
4) Suppose that the eigensystem of a 2 £ 2 system A is ¸1 Ã!
De…ne D and S as in (2).
(a) Calculate AS
(b) Calculate SD:
(c) Conclusion:
(Ã
®
¯
!)
; ¸2 Ã!
(Ã
°
±
!)
: