Linear algebra: Jordan forms & eigenvectors

**bkarpuz** · May 2nd 2011, 08:30 PM

Dear MHF members,

I have the following problem.
If the Jordan form of $A$ is a single Jordan block of size $n \times n$ ,
then how many independent eigenvectors does A have and what is the dimension of $\ker (A-\lambda I)^2$ ?

Thanks for your help.
bkarpuz

**tonio** · May 3rd 2011, 03:57 AM

Originally Posted by bkarpuz

Dear MHF members,

I have the following problem.
If the Jordan form of $A$ is a single Jordan block of size $n \times n$ ,
then how many independent eigenvectors does A have and what is the dimension of $\ker (A-\lambda I)^2$ ?

Thanks for your help.
bkarpuz

As the number of Jordan blocks corr. to an eigenvalue equals the dimension of the eigenspace corr. to that eigenvalue, we have in this

case that there's one unique eigenvalue for that matrix and its eigenspace is of dimension 1...

Tonio

**FernandoRevilla** · May 3rd 2011, 05:14 AM

Originally Posted by bkarpuz

and what is the dimension of $\ker (A-\lambda I)^2$ ?

One way for this question (not the most elegant): if J is the Jordan form of A, then

$(J-\lambda I)^2=\begin{bmatrix} 0 & 1 & 0 & & \ldots &0 & 0 & 0\\ 0 & 0 & 1 & &\ldots &0 & 0 & 0 \\0 & 0 & 0 & & \ldots & 0 & 0 & 0\\ \vdots&&&&&&&\vdots \\ 0 & 0 & 0 & & \ldots & 0 & 1 & 0\\ 0 & 0 & 0 & & \ldots & 0 & 0 & 1\\0 & 0 & 0 & &\ldots & 0 & 0 & 0\end{bmatrix}^2=\begin{bmatrix} 0 & 0 & 1 & & \ldots &0 & 0 & 0\\ 0 & 0 & 0 & &\ldots &0 & 0 & 0 \\0 & 0 & 0 & & \ldots & 0 & 0 & 0\\ \vdots&&&&&&&\vdots \\ 0 & 0 & 0 & & \ldots & 0 & 0 & 1\\ 0 & 0 & 0 & & \ldots & 0 & 0 & 0\\0 & 0 & 0 & &\ldots & 0 & 0 & 0\end{bmatrix}$

As a consequence,

$\dim \ker (A-\lambda I)^2=\dim \ker (J-\lambda I)^2=n-\textrm{rank}(J-\lambda I)^2=n-(n-2)=2$

Thread: Linear algebra: Jordan forms & eigenvectors

LinkBack

Thread Tools

Display

Linear algebra: Jordan forms & eigenvectors

Similar Math Help Forum Discussions

Eigenvalues and eigenvectors [Linear Algebra]

[SOLVED] Linear Algebra, basis of eigenvectors

Linear Algebra, eigenvectors problem

linear algebra with transformations and eigenvectors

Linear Algebra Eigenvectors -- question on x1+x2+x3 = 0 and more

Search Tags