- #1
bad throwaway name
- 6
- 0
Homework Statement
For each matrix A, I need to find a basis for each generalized eigenspace of ## L_A ## consisting of a union of disjoint cycles of generalized eigenvectors. Then I need to find the Jordan canonical form of A.
The matrices are:
## a)
\begin{pmatrix}
1 & 1\\
-1 & 3
\end{pmatrix}
b)
\begin{pmatrix}
1 & 2\\
3 & 2
\end{pmatrix}
c)
\begin{pmatrix}
11 & -4 & -5\\
21 & -8 & -11\\
3 & -1 & 0
\end{pmatrix}
d)
\begin{pmatrix}
2 & 1 & & \\
& 2 & 1 & \\
& & 3 & \\
& 1 & -1 & 3
\end{pmatrix} ##
Homework Equations
## P_A(t)=det(A-tI) ##
## K_\lambda = \left \{ (A-\lambda I)^j \right \}, 1\leq j \leq p ## where p is the minimum value for which ## (A-\lambda I)^j(x)=0## for a generalized eigenvector x
The Attempt at a Solution
I hope you don't mind if I link to a picture of what I have so far, as I don't want to go through the trouble of typing it all up.
http://prntscr.com/am7si6
The squiggles next to some of the matrices are just messily written words indicating that there is only one generalized eigenvector for a given basis.
For (a), I determined that the only basis element was (1,1)
For (b), I found 2 single-element basis which contained (1,-1) and (1,1.5)
For (c), I once again found that each generalized eigenspace only had one generalized eigenvector. However, what I'm not sure of is how to determine what the Jordan blocks are in this case- i.e, I don't know how to determine if
## J=\begin{pmatrix}
-1 & 1 & \\
& -1 & \\
& & 2
\end{pmatrix} ## or
##J=\begin{pmatrix}
-1 & & \\
& 2 & 1\\
& & 2
\end{pmatrix}##
For (d), I have not yet started. I just need to confirm that my current approach of raising the power of the generalized matrices to find generalized eigenvectors is correct, and I need to know how to figure out what the Jordan blocks are.