MATLAB Lab for Math 526 Lab J
Eigenvalues and Eigenvectors
Department of Mathematics
Overview
Use the MATLAB tools for finding eigenvalues and eigenvectors.
Activities
For a square matrix A,
d=eig(A)
returns a vector that contains the eigenvalues of A.
[S, Lambda]=eig(A)
returns the diagonal matrix Lambda with the eigenvalues of A. The columns of the
matrix Srepresent the corresponding eigenvectors, but they have been normalized so
that each column represents a unit vector.
Test this by finding the eigenvalues and their corresponding eigenvectors of A=
1 1 −1 0
0 2 1 0
0−310
4 5 −3 2
, and then check them by computing Axand λxfor each of the four
sets of eigenvalues and eigenvectors. In this, you will also see how Matlab handles
complex numbers.
Assignments (Due 5pm March 6, Friday)
If A=1
4
492 83 −214 0 46 9
−290 −36 108 −12 −21 10
970 164 −422 0 91 18
216 51 −114 −14 27 21
−450 −100 232 24 −53 −38
1662 293 −740 −12 161 45
, limk→∞ Ak
−3
3
−3
−1
15
−12
=
0
2
0
2
−4
2
,
limk→∞ Ak
3
4
7
12
−10
18
=
0
0
0
0
0
0
, and limk→∞ Ak
−2
4
−1
2
11
−7
does not exist. (In the last one, four
of the entries go to plus or minus infinity, but two approach finite values.) It can be
difficult to estimate these limits by telling Matlab to compute Akbfor large values of k
because of the accumulated round-off error in applying Aseveral times. Indeed, for the
second limit, when kis greater than about 25, applying Aagain makes the vector move
April 2, 2009
