122 CHAPTER 6. DIRECT SUMS AND BLOCK DIAGONAL MATRICES

where a is a 1× 1 matrix. Thus a = 2 and so the matrix of A with respect to the orderedbasis given above is  1 0 0

0 1 00 0 2

What if you changed the order of the vectors in the basis? Suppose you had them orderedas 

 110

 ,

 0−12

 ,

 101



Then you would have three invariant subspaces whose direct sum is R3,

span

 1

10

 ,span

 0−12

 , and span

 1

01



Then the matrix of A with respect to this ordered basis is 1 0 00 2 00 0 1

Example 6.0.6 Consider the following matrix.

A =

 3 1 0−1 1 0−1 −1 1

Let

V1 ≡ span

 0

01

 ,V2 ≡ span

 1

0−1

 ,

 1−10



Show that these are A invariant subspaces and find the matrix of A with respect to theordered basis 

 001

 ,

 1−10

 ,

 10−1



First note that 3 1 0−1 1 0−1 −1 1

−2

 1 0 00 1 00 0 1

 1

0−1

=

 1−10

