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Chapter 10. Trace and Determinant
246
Exercise 19 fails on
infinite-dimensional
if
∗
inner-product spaces, 19. Suppose V is an inner-product space and T ∈L(V). Prove that
T v ≤ Tv
leading to what are
for every v ∈ V, then T is normal.
called hyponormal
operators, which have a 20. Prove or give a counterexample: if T ∈L(V) and c ∈ F, then
well-developed theory. det(cT) = c dim V det T.
21. Prove or give a counterexample: if S, T ∈L(V), then det(S+T) =
det S + det T.
22. Suppose A is a block upper-triangular matrix
A 1 ∗
.
. ,
A = .
0 A m
where each A j along the diagonal is a square matrix. Prove that
det A = (det A 1 )...(det A m ).
n
23. Suppose A is an n-by-n matrix with real entries. Let S ∈L(C )
n
denote the operator on C whose matrix equals A, and let T ∈
n
n
L(R ) denote the operator on R whose matrix equals A. Prove
that trace S = trace T and det S = det T.
24. Suppose V is an inner-product space and T ∈L(V). Prove that
det T ∗ = det T.
√
Use this to prove that |det T|= det T T, giving a different
∗
proof than was given in 10.37.
25. Let a, b, c be positive numbers. Find the volume of the ellipsoid
$ 3 x 2 y 2 z 2 %
(x,y,z) ∈ R : + + < 1
a 2 b 2 c 2
3
by finding a set Ω ⊂ R whose volume you know and an operator
3
T ∈L(R ) such that T(Ω) equals the ellipsoid above.