138 CHAPTER 6. DIRECT SUMS AND BLOCK DIAGONAL MATRICES

which says that solutions are of the form 3zyz

 ,y,z ∈ R not both 0

This is the nonzero vectors of.

span

 3

01

 ,

 010



Note these are not eigenvectors. They are called generalized eigenvectors because theypertain to ker

((A−2I)2

)rather than ker((A−2I)) . What is the matrix of the restriction

of A to this subspace having ordered basis 3

01

 ,

 010

 ,

 2−11



A

 301

 =

 10 12 −6−4 −4 33 4 −1

 3

01

=

 24−98

A

 010

 =

 10 12 −6−4 −4 33 4 −1

 0

10

=

 12−44

Then  24 12

−9 −48 4

=

 3 00 11 0

M (6.1)

and so some computations yield

M =

(8 4−9 −4

)Indeed this works  3 0

0 11 0

( 8 4−9 −4

)=

 24 12−9 −48 4

Then the matrix associated with the other eigenvector is just 1. Hence the matrix withrespect to the above ordered basis is 8 4 0

−9 −4 00 0 1

