Page 206 - Linear Algebra Done Right

Chapter 9. Operators on Real Vector Spaces
196
10
11


14
 0
25 


−3
0
−3
17
16


A =  4 −3 −3 12 13 
 
 0 0 0 5 5 
 
0 0 0 5 5
is a block upper-triangular matrix with
 
A 1 ∗
 
 ,
A =  A 2
0 A 3
where

−3 −3 5 5
A 1 = 4 , A 2 = , A 3 = .
−3 −3 5 5
Every upper-triangular Now we prove that for each operator on a real vector space, we can
matrix is also a block find a basis that gives a block upper-triangular matrix with blocks of
upper-triangular matrix size at most 2-by-2 on the diagonal.
with blocks of size
1-by-1 along the 9.4 Theorem: Suppose V is a real vector space and T ∈L(V).
diagonal. At the other Then there is a basis of V with respect to which T has a block upper-
extreme, every square triangular matrix
matrix is a block
 
upper-triangular matrix A 1 ∗
 . 
because we can take 9.5  . .  ,
 
the first (and only) 0 A m
block to be the entire
matrix. Smaller blocks where each A j is a 1-by-1 matrix or a 2-by-2 matrix with no eigenvalues.
are better in the sense
that the matrix then Proof: Clearly the desired result holds if dim V = 1.
has more 0’s. Next, consider the case where dim V = 2. If T has an eigenvalue λ,
then let v 1 ∈ V be any nonzero eigenvector. Extend (v 1 ) to a basis
(v 1 ,v 2 ) of V. With respect to this basis, T has an upper-triangular
matrix of the form

λ a
0 b .
In particular, if T has an eigenvalue, then there is a basis of V with
respect to which T has an upper-triangular matrix. If T has no eigen-
values, then choose any basis (v 1 ,v 2 ) of V. With respect to this basis,