Page 100 - Linear Algebra Done Right

Diagonal Matrices
Proof: Suppose (v 1 ,...,v n ) is a basis of V with respect to which
T has an upper-triangular matrix

λ 1 ∗  87
 
 λ 2 


  .
M T, (v 1 ,...,v n ) =  . .
 . 
 
0 λ n
Let λ ∈ F. Then
 
λ 1 − λ ∗
 
 λ 2 − λ 

 
M T − λI, (v 1 ,...,v n ) =  . .  .
 . 
 
0 λ n − λ
Hence T − λI is not invertible if and only if λ equals one of the λ s

j
(see 5.16). In other words, λ is an eigenvalue of T if and only if λ

equals one of the λ s, as desired.
j
Diagonal Matrices
A diagonal matrix is a square matrix that is 0 everywhere except
possibly along the diagonal. For example,
 
8 0 0


 0 2 0 
0 0 5
is a diagonal matrix. Obviously every diagonal matrix is upper triangu-
lar, although in general a diagonal matrix has many more 0’s than an
upper-triangular matrix.
An operator T ∈L(V) has a diagonal matrix
 
λ 1 0
 . 
 . . 
 
0 λ n
with respect to a basis (v 1 ,...,v n ) of V if and only
Tv 1 = λ 1 v 1
. . .

Tv n = λ n v n ;