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(Solved): Please show all work in finding the vectors. C: An example of diagonalisation : Let T be the operat ...



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Please show all work in finding the vectors.

An example of diagonalisation : Let be the operator of with matrix 1) Find 3 vectors satisfying the equations : 2) Find the matrix of in the basis . 3) Use this to find with as little computation as possible.


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Now we can see that given matrix M is a upper triangular matrix so its diagonal elements 1,i,-i are its eigenvalues.

1)
Now   
   where    is identity matrix.
  
  
Now writing this system in augmented matrix we get
  
Now we find row reduced echelon form
Multiply each element of    by    to make the entry at    a   .
  

Perform the row operation   
  

Multiply each element of    by   
  
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