Block Diagonal Matrix and Similarity Transformation

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SUMMARY

The discussion focuses on the transformation matrix, ν, in the context of block diagonal matrices and similarity transformations. The user references diagonalization, where D = S-1MS, and contrasts it with the Jordan Form, which is applicable when matrices cannot be fully diagonalized. The user clarifies that the Jordan Form consists of upper-triangular blocks, indicating that the notes referenced do not adhere to this structure.

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nigelscott
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I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemist...stry-ii-fall-2008/lecture-notes/Lecture_3.pdf

How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S are the eigenvectors of M but this doesn't appear to be the case here.

Any help would be appreciated.
 
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Some matrices cannot be diagonalized. In such cases the "best" you can do is to make them block-diagonal; in linear algebra this is called the Jordan Form. To me it looks like this is what the notes are talking about.

EDIT: right after posting, I remembered that the Jordan form has upper-triangular blocks, so the notes are not doing that...

Jason
 

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