Page 96 - Linear Algebra Done Right

Upper-Triangular Matrices
and let v be a corresponding nonzero eigenvector. Extend (v) to a
basis of V. Then the matrix of T with respect to this basis has the form
above. Soon we will see that we can choose a basis of V with respect to 83
which the matrix of T has even more 0’s.
The diagonal of a square matrix consists of the entries along the
straight line from the upper left corner to the bottom right corner.
For example, the diagonal of the matrix 5.11 consists of the entries
a 1,1 ,a 2,2 ,...,a n,n .
A matrix is called upper triangular if all the entries below the di-
agonal equal 0. For example, the 4-by-4 matrix
 
6 2 7 5
 
 0 6 1 3 
 
 0 0 7 9 


0 0 0 8
is upper triangular. Typically we represent an upper-triangular matrix
in the form
 
λ 1 ∗
 . 
 . .  ;
 
0 λ n
the 0 in the matrix above indicates that all entries below the diagonal
in this n-by-n matrix equal 0. Upper-triangular matrices can be consid-
ered reasonably simple—for n large, an n-by-n upper-triangular matrix
has almost half its entries equal to 0.
The following proposition demonstrates a useful connection be-
tween upper-triangular matrices and invariant subspaces.

5.12 Proposition: Suppose T ∈L(V) and (v 1 ,...,v n ) is a basis
of V. Then the following are equivalent:

(a) the matrix of T with respect to (v 1 ,...,v n ) is upper triangular;
(b) Tv k ∈ span(v 1 ,...,v k ) for each k = 1,...,n;
(c) span(v 1 ,...,v k ) is invariant under T for each k = 1,...,n.

Proof: The equivalence of (a) and (b) follows easily from the def-
initions and a moment’s thought. Obviously (c) implies (b). Thus to
complete the proof, we need only prove that (b) implies (c). So suppose
that (b) holds. Fix k ∈{1,...,n}. From (b), we know that