Eigenvectors and Eigenvalues

Wednesday, 5 September 2018

2:15 PM

The eigenvectors  of  for  are the non-zero solutions to

Which is

 

Reduce the augmented matrix with the right hand zero column omitted

 

Hence

Put , then we obtain

The eigenvectors are ,

 

 

 

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Let  be an  matrix

Suppose that we have , non-zero

 

Then  is an eigenvalue with corresponding eigenvector

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To find the eigenvalues of A, find the zeroes of the characteristic polynomial

For each eigenvalue, the eigenvectors are the non-zero vectors in

 

A matrix  is diagonalisable if there is a diagonal matrix  and invertible matrix  such that

 

An  matrix  is diagonalisable if it has  linearly independent eigenvectors