Nearest block diagonal matrix to a given matrix

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SUMMARY

The discussion centers on the problem of finding the nearest block diagonal matrix to a given matrix, specifically in the context of minimizing the Frobenius norm. The user initially sought assistance for a quantum chemistry manuscript but ultimately resolved the issue independently after a week of contemplation. The uniqueness of the block diagonal form, defined by unitary rotations of the diagonal blocks, is acknowledged as a key aspect of the problem.

PREREQUISITES
  • Understanding of matrix theory, particularly block diagonal matrices
  • Familiarity with the Frobenius norm and its applications
  • Knowledge of unitary transformations in linear algebra
  • Basic concepts in quantum chemistry related to matrix representations
NEXT STEPS
  • Research methods for computing the nearest block diagonal matrix
  • Learn about the Frobenius norm and its properties in matrix analysis
  • Explore unitary transformations and their implications in linear algebra
  • Investigate applications of block diagonal matrices in quantum chemistry
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Researchers in quantum chemistry, mathematicians specializing in linear algebra, and anyone involved in matrix optimization problems.

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Suppose I have a matrix that I want to reduce to block diagonal form. Obviously, the block diagonal form is not unique as each of the diagonal blocks is defined only to within a unitary rotation. So I want to find the block diagonal matrix that is closest to the original matrix in terms of the Frobenius norm.

This problem arises in a quantum chemistry manuscript I am putting together. If anyone can point me to a solution for this, I would be more than happy to acknowledge them in the manuscript.

Thanks!
 
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Okay, I figured it out myself. After thinking about it for a week, I solve it right after posting!
 

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