- #1

fluidistic

Gold Member

- 3,767

- 136

Let's say I have a 6x6 matrix A whose Jordan form J has 3 Jordan blocks. It means that this matrix (matrix A, but I think that also the matrix J) has 3 linearly independent eigenvectors, I have no problem in finding them. I simply do [itex](A-\lambda _i I)v_i=0[/itex] to get the eigenvectors [itex]v_i[/itex].

Now when I want to find the matrix S such that [itex]A=SJS^{-1}[/itex], I already know half of the matrix S. More precisely the 3 eigenvectors I have are 3 of the column vectors of the matrix S. Now I need to complete the matrix S with other 3 linearly independent vectors (but not eigenvectors!) but I have no idea how to do this.

Here is the particular example where wolfram found out those 3 vectors, but even by looking at them I have no idea how to find them. http://www.wolframalpha.com/input/?i=jordan+form+{{2%2C1%2C-1%2C0%2C1%2C0}%2C{0%2C2%2C0%2C1%2C1%2C0}%2C{0%2C0%2C2%2C0%2C1%2C0}%2C{0%2C0%2C0%2C2%2C1%2C0}%2C{0%2C0%2C0%2C0%2C2%2C1}%2C{0%2C0%2C0%2C0%2C0%2C-1}}

Any idea on how to get those 3 column vectors in the matrix S?