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I've wondered whether someone could calrify me what are Generalized Eigenvectors, and why can I use them to find triangular form of a matrix.
Say I have a $3\times3$ matrix, and I want to bring it to its triangular form. Say I find one Eigenvalue of algebric multiplicity 3, and Eigenvector of geometric multiplicity 1.
Why will solving $(A-\lambda I)v2 = v1$ (and getting Generalized Eigenvectors as someone did here) give me more vectors, that combining with the first eigenvector I had found, will make an invertible matrix that I can use to triangle the given matrix? Why does it work? And will it always work? And do I have another find to make this matrix triangular one?
Why does eigenvector of geometric multiplicity 3 will produce me a diagonal matrix, but Generalized Vectors a triangular one? I've tried searching for the answers myself but didn't find anything useful. If someone could explain or point me to a good source that'll be great.

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  • $\begingroup$ I did, and didn't find a direct or satisfying answes to my questions. I might didn't understand it (though I don't believe so), but of course I've tried wikipedia before opening this post. I'm looking for a simple explanation or another source $\endgroup$ – user114138 Dec 12 '14 at 11:37
  • $\begingroup$ Do you understand Jordan canonical form? $\endgroup$ – Omnomnomnom Dec 12 '14 at 18:59

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