Question

Let (V, C) be a finite-dimensional complex inner product space.

We recall that a map T : V → V is said to be normal if
T^{∗} ◦ T = T ◦ T^{∗} .

1. Show that if T is normal, then |T^{∗}(v)| = |T(v)|
for all vectors v ∈ V.

2. Let T be normal. Show that if v is an eigenvector of T
relative to the eigenvalue λ, then it is also an eigenvector of
T^{∗} relative to the eigenvalue λ.

3. When is the normal operator T self-adjoint? Motivate your answer!

Answer #1

Say (V,<,>) is a finite dimensional real inner product
space. Then the composition of two self adjoint operators is again
self adjoint. Prove or Disprove.

Let V be a finite dimensional vector space over R with an inner
product 〈x, y〉 ∈ R for x, y ∈ V .
(a) (3points) Let λ∈R with λ>0. Show that
〈x,y〉′ = λ〈x,y〉, for x,y ∈ V,
(b) (2 points) Let T : V → V be a linear operator, such that
〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V.
Show that T is one-to-one.
(c) (2 points) Recall that the norm of a vector x ∈ V...

Suppose that V is a finite dimensional inner product space over
C and dim V = n, let T be a normal linear transformation of V
If S is a linear transformation of V and T has n distinc
eigenvalues such that ST=TS. Prove S is normal.

Let f be a non-degenerate form on a finite-dimensional space V. Show that
each linear operator S has an adjoint relative to f.

Let V be a finite-dimensional vector space and let T be a linear
map in L(V, V ). Suppose that dim(range(T 2 )) = dim(range(T)).
Prove that the range and null space of T have only the zero vector
in common

Let V be a finite dimensional space over R, with a positive
definite scalar product. Let P : V → V be a linear map such that P
P = P . Assume that P T P = P P T . Show that P = P T .

Let V be a finite-dimensional vector space over C and T in L(V)
be an invertible operator in V. Suppose also that T=SR is the polar
decomposition of T where S is the correspondIng isometry and
R=(T*T)^1/2 is the unique positive square root of T*T. Prove that R
is an invertible operator that committees with T, that is
TR-RT.

(3) Let V be a finite dimensional vector space, and let T: V® V
be a linear transformation such that rk(T) = rk(T2).
a) Show that ker(T) = ker(T2).
b) Show that 0 is the only vector that lies in
both the null space of T, and the range space of T

Suppose V is a ﬁnite dimensional inner product space. Prove that
every orthogonal operator on V , i.e. <T(u), T(v)> , ∀u,v ∈ V
, is an isomorphism.

Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be a set of inner products on V that
is (when viewed as a subset of B(V)) linearly independent. Prove
that S must be finite
e) Exhibit an infinite linearly independent set of inner
products on R(x), the vector space of all polynomials with real
coefficients.

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