Page 254 - Linear Algebra Done Right

Exercises
3
12.
Suppose T ∈L(C ) is the operator whose matrix is

51

−40

 60 −12 −21  245
−28  .
57 −68 1
Someone tells you (accurately) that −48 and 24 are eigenvalues
of T. Without using a computer or writing anything down, find
the third eigenvalue of T.
13. Prove or give a counterexample: if T ∈L(V) and c ∈ F, then
trace(cT) = c trace T.
14. Prove or give a counterexample: if S, T ∈L(V), then trace(ST) =
(trace S)(trace T).
15. Suppose T ∈L(V). Prove that if trace(ST) = 0 for all S ∈L(V),
then T = 0.
16. Suppose V is an inner-product space and T ∈L(V). Prove that
if (e 1 ,...,e n ) is an orthonormal basis of V, then
2
2
trace(T T) = Te 1 +· · ·+HTe n .
∗
Conclude that the right side of the equation above is independent
of which orthonormal basis (e 1 ,...,e n ) is chosen for V.
17. Suppose V is a complex inner-product space and T ∈L(V). Let
λ 1 ,...,λ n be the eigenvalues of T, repeated according to multi-
plicity. Suppose
 
a 1,1 ... a 1,n
 . . 
 . . . . 
 
a n,1 ... a n,n
is the matrix of T with respect to some orthonormal basis of V.
Prove that
n n

2 2 2
|λ 1 | +· · ·+|λ n | ≤ |a j,k | .
k=1 j=1
18. Suppose V is an inner-product space. Prove that
S, T = trace(ST )
∗
defines an inner product on L(V).