Find the Jordan Canonical form of the following matrice:
For this one, here is what I have so far:
First, I found the eigenvalues are 1 and 2, and the basis of eigenvectors are
Now, I know I have to find , but I'm bit rusty on this, I can't remember how to do it, any hint, please?
Thanks.
The rule is that if an eigenvalue λ occurs with multiplicity 2, you have to find the dimension of the corresponding eigenspace (in other words, how many linearly independent eigenvectors are there for that eignvalue). The dimension of the eigenspace will be either 1 or 2. If it is 2, then the Jordan canonical form corresponding to that eigenvalue just consists of two λs on the diagonal. If it is 1 then the canonical form has a 2×2 block .
Fot the above matrix A, each eigenvalue has an eigenspace of dimension 1 (an eigenvector for the eigenvalue 2 must be a multiple of (1,0,0,0) and an eigenvector for the eigenvalue 3 must be a multiple of (0,0,0,1)). So the JCF looks like .
Thanks, but I cannot understand why the the eigenvectors are such, let me show you how I worked mine, please correct me.
For a)
The eigenvalues are -1 and 4, so the eigenspace for -1 is:
So I have to solve
Let , we have , so the eigenvectors for this is
But appearly it is not, what am I doing wrong here?