How to determine which eigenvalue has multiplicity 2 ?

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SUMMARY

The discussion focuses on determining which eigenvalue of a 4x4 matrix has multiplicity 2 without resorting to the characteristic polynomial or row reducing matrices. The user proposes using the trace of the matrix, defined as the sum of the diagonal elements, to find the repeated eigenvalue. By setting up the equation Ʃ(elements of diagonal) = λ1 + λ2 + λ3 + x, where x represents the unknown eigenvalue, the user can easily calculate the multiplicity without extensive computations.

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sid9221
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Say I have a 4x4 matrix and I know 3 eigenvalues and the 3 corresponding eigenvectors.

Is there a fast way to calculate which one has multiplicity 2 without calculating the characteristic polynomial(too time consuming for a 4x4 matrix) or without determining the dimensions of (A-λ I) for each eigenvalue λ. (Row reducing 3 4x4 matrices is too long during an exam)
 
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LOL i just though of the solution as soon as I typed the question.

I guess I could set up an equation using the definition of trace:

Ʃ(elements of diagnal)=λ1+λ2+λ3+x

hence work out x, which will tell me which eigenvalue is repeated.
 
nice! :smile:
 

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