problems4.1 - Math 307: Problems for section 4.1 November...

Math 307: Problems for section 4.1
November 14, 2012
1. For the following matrices find
(a) all eigenvalues
(b) linearly independent eigenvectors for each eigenvalue
(c) the algebraic and geometric multiplicity for each eigenvalue
and state whether the matrix is diagonalizable.
A =
bracketleftbigg
3 7
2 - 2
bracketrightbigg
(calculate by hand)
B =


1 - 3 3
3 - 5 3
6 - 6 4


(calculate using Matlab/Octave or otherwise)
C =


1 2 1
2 0 - 2
- 1 2 3


(calculate using Matlab/Octave or otherwise)
2. Find a 3 × 3 real, non-zero ( i.e. not all entries zero) matrix which has all three eigenvalues
zero.
3. (a) By hand find a matrix with eigenvalues λ
1
= 1 and λ
2
= 2 and corresponding eigen-
vectors
v
1
=
bracketleftbigg
1
2
bracketrightbigg
v
2
=
bracketleftbigg
2
1
bracketrightbigg
(b) Using Matlab/Octave or otherwise, find a matrix with eigenvalues