Finding unitary transformation

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SUMMARY

The discussion focuses on finding a unitary transformation that diagonalizes the matrix given by the user. The matrix in question is a 4x4 matrix with repeated rows, indicating linear dependence. The user suggests that instead of calculating the determinant of the entire matrix, one can reduce the problem by identifying eigenvalues and eigenvectors through inspection. The conversation highlights the challenge of guessing the remaining eigenvectors after identifying two easily found ones.

PREREQUISITES
  • Understanding of unitary transformations
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with matrix diagonalization
  • Basic skills in linear algebra
NEXT STEPS
  • Study the process of finding eigenvalues for 4x4 matrices
  • Learn techniques for matrix reduction to simplify calculations
  • Explore methods for guessing eigenvectors in linear algebra
  • Investigate unitary matrices and their properties in detail
USEFUL FOR

Students studying linear algebra, mathematicians working on matrix theory, and anyone interested in the application of unitary transformations in quantum mechanics or advanced mathematics.

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Homework Statement



Find a unitary transformation that diagonalizes the matrix:

1 1 1 -3
1 1 1 -3
1 1 1 -3
-3 -3 -3 -9


Homework Equations





The Attempt at a Solution


So before I even start with finding the eigenvalues for this, I know there has to be a way to reduce this so I don't have to find the determinant of a 4x4 matrix. Clearly none of these rows are independent. We haven't gone over this very thoroughly in class, so I'm not too sure about the best way to go about this.
 
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There's no real shortcut here. Sometimes you can find eigenvectors by inspection and guessing. Two of them are pretty easy to find. I'm not how you would guess the other two, unless you are better at guessing than I am. Just evaluate the determinant.
 

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