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estro
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In my book I see that the author finds the jordan basis for matrix A={(-3,9),(1,3)} by finding nullspace for (A-3I) and (A-3I)^2 without any justification, do I miss something trivial here?
estro said:In my book I see that the author finds the jordan basis for matrix A={(-3,9),(1,3)} by finding nullspace for (A-3I) and (A-3I)^2 without any justification, do I miss something trivial here?
estro said:Thank you! I should have been looking there in the first place=...)
If you don't mind I'll ask here again if wikipedia won't be enough for me.
estro said:After reading the wiki and the proof for the existence of the jordan form I think that I'm getting into the idea, however I was able to think about the following example:
Lets choose matrix A={(2,0,0),(0,0,1),(0,0,0)} so the characteristic polynomial is also the minimal: p(t)=(t-2)t^2.
Now this is how I find the jordan basis:
1. NullSpace(A-2I)=Sp{(1,0,0)}
2. NullSpace(A-0I)=SP{(0,1,0)}
3. NullSpace(A-0I)^2={(0,0,1),(0,1,0)}
So the jordan basis is {(1,0,0),(0,1,0),(0,0,1)}
But how can I in what order to write these column vectors in my matrix P? [to satisfy P^{-1}AP]
estro said:Thanks!
But did I understand the concept of finding jordan basis? []
Will I be able to find the jordan basis for every possible matrix with this technique?
The Jordan Basis for Matrix A is a set of vectors that spans the entire space of a given matrix A. These vectors are the generalized eigenvectors of A and can be used to diagonalize A.
The Jordan Basis for Matrix A is calculated by finding the eigenvalues and eigenvectors of A and then finding the generalized eigenvectors by solving a system of equations.
The Jordan Basis for Matrix A is important because it allows us to easily find the Jordan form of A, which is useful for solving systems of linear equations and understanding the behavior of A.
Yes, the Jordan Basis for Matrix A can be used for any square matrix. However, it is most useful for matrices with repeated eigenvalues.
If the Jordan Basis for Matrix A is missing a trivial vector, it means that A is not diagonalizable. This could happen if A has a repeated eigenvalue but not enough generalized eigenvectors to span the space. In this case, the Jordan form of A will have blocks of size larger than 1 along the diagonal.