Page 220 - Linear Algebra Done Right

Chapter 9. Operators on Real Vector Spaces
210
Exercises
1. Prove that 1 is an eigenvalue of every square matrix with the
property that the sum of the entries in each row equals 1.
2. Consider a 2-by-2 matrix of real numbers

a c
A = .
b d
Prove that A has an eigenvalue (in R) if and only if
2
(a − d) + 4bc ≥ 0.
3. Suppose A is a block diagonal matrix
 
A 1 0
 . 
A =  . .  ,
 
0 A m
where each A j is a square matrix. Prove that the set of eigenval-
ues of A equals the union of the eigenvalues of A 1 ,...,A m .

Clearly Exercise 4 is a 4. Suppose A is a block upper-triangular matrix
stronger statement  
A 1 ∗
than Exercise 3. Even  . . 
so, you may want to do A =  .  ,


Exercise 3 first because 0 A m
it is easier than where each A j is a square matrix. Prove that the set of eigenval-
Exercise 4. ues of A equals the union of the eigenvalues of A 1 ,...,A m .

5. Suppose V is a real vector space and T ∈L(V). Suppose α, β ∈ R
2
are such that T + αT + βI = 0. Prove that T has an eigenvalue
2
if and only if α ≥ 4β.
6. Suppose V is a real inner-product space and T ∈L(V). Prove
that there is an orthonormal basis of V with respect to which T
has a block upper-triangular matrix
 
A 1 ∗
 . 
 . .  ,
 
0 A m
where each A j is a 1-by-1 matrix or a 2-by-2 matrix with no eigen-
values.