Page 154 - Linear Algebra Done Right

Chapter 7. Operators on Inner-Product Spaces
142
To prove (e), note that in (d) we showed that the restriction of T to
any invariant subspace is normal. However, U
(by (a)), and hence T| U ⊥ is normal. ⊥ is invariant under T
In proving 7.18 we thought of a matrix as composed of smaller ma-
trices. Now we need to make additional use of that idea. A block diag-
The key step in the onal matrix is a square matrix of the form
proof of the last  
A 1 0
proposition was
 . 
 .  ,
showing that M(T) is  . 
an appropriate block 0 A m
diagonal matrix;
where A 1 ,...,A m are square matrices lying along the diagonal and all
see 7.21.
the other entries of the matrix equal 0. For example, the matrix
 
4 0 0 0 0
 0 2 −3 0 0 


 
7.22 A =  0 3 2 0 0 
 

 0 0 0 1 −7 

0 0 0 7 1
is a block diagonal matrix with
 
A 1 0
 
A =  A 2  ,
0 A 3
where

2 −3 1 −7
7.23 A 1 = 4 , A 2 = , A 3 = .
3 2 7 1
If A and B are block diagonal matrices of the form
   
A 1 0 B 1 0
 .   . 
A =  . .  , B =  . .  ,
   
0 A m 0 B m
where A j has the same size as B j for j = 1,...,m, then AB is a block
diagonal matrix of the form
 
A 1 B 1 0
 . 
7.24 AB =  . .  ,
 
0 A m B m