• Support PF! Buy your school textbooks, materials and every day products Here!

Diagonalizable matrix problem with parameters

  • Thread starter ryebread
  • Start date
  • #1
2
0

Homework Statement


A is a given 3x3 matrix
A=[tex]
\begin{pmatrix}
a & 0 & 1 \\
0 & a & b \\
0 & 0 & c
\end{pmatrix}
[/tex]
a,b,c are real numbers
1) determine for what values of a,b,c the matrix A is diagnoizable. (advice: distinguish between the cases a=c and a≠c)

Homework Equations


characteristic polynomial of A is det(tI-A)

The Attempt at a Solution



finding the solution for det(tI-A)=0
tI-A=[tex]
\begin{pmatrix}
t-a & 0 & -1 \\
0 & t-a & -b \\
0 & 0 & t-c
\end{pmatrix}
[/tex]
the characteristic polynomial is (t-c)(t-a)^2 and therefore the eigenvalues are a and c.
for t=c we get the matrix [tex]
\begin{pmatrix}
c-a & 0 & -1 \\
0 & c-a & -b \\
0 & 0 & 0
\end{pmatrix}
[/tex]
let it be C, x=(x,y,z), for Cx=0 we get the system of linear equations:
(c-a)x-z=0
(c-a)y -bz=0
likewise, for t=a we get
-1z=0
-bz=0
(a-c)z=0

after that I'm pretty much lost. my guess is that you need to find the dimension of the eigenspace that corresponds to each eigenvalue, but I'm nut sure how.

Help would be very welcomed
 

Answers and Replies

  • #2
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,394
179
As the problem suggests, consider the a=c case separately. If a=c, then your equations above simplify to
-z = 0
-bz = 0

What constraints does this impose on x, y, and z?
 
  • #3
2
0
As the problem suggests, consider the a=c case separately. If a=c, then your equations above simplify to
-z = 0
-bz = 0

What constraints does this impose on x, y, and z?
i have no idea.
I finally managed to solve it with algebraic and geometric multiplicities, but i just don't get nullity.
 

Related Threads for: Diagonalizable matrix problem with parameters

  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
568
  • Last Post
2
Replies
33
Views
3K
Top