Chapter 4 - Inner product Spaces

Question 1

Definition 1.1 : Inner product space

Answer 1

An inner product space is a pair (V, < . , . >), where V is a vector space and < . , . > : V x V -> K(bar) , (u,v) |-> <u,v>, satisfies the following axioms:
i) < u+v, w > = < u, w > + < v, w > (u,v,w ϵ V)
ii) < αv, w > = α < v, w > (α ϵ K(bar) , v,w ϵ V)
iii) < v, w > = < w, v > (with a bar ontop denoting the complex conjugate) (v,w ϵ V)
iv) < v, v > >0 (v ϵ V \ {0})

Question 2

Definition 1.5 : Linear isometry

Answer 2

Let V and W be inner product spaces. A linear map T:V->W that satisfies ||Tv|| = ||v|| (v ϵ V) is called a linear isometry. A bijective linear isometry is called an isomorphic isometry. 2 inner product spaces are said to be isometrically isomorphic if there exists an isometric isomorphism between them.

Question 3

Definition 2.1 : Orthogonal vectors

Answer 3

Two vectors v and w in an inner product space are said to be orthogonal if < v, w > = 0. If, in addition, ||v||=||w||= 1 then they are said to be orthonormal. A set of mutually orthogonal vectors in an inner product space is a set in which any two vectors are orthogonal. A set of mutually orthogonal vectors of length 1 is called an orthonormal set.

Question 4

Definition 3.1 : Adjoint of a linear map

Answer 4

Let V and W be inner product spaces over C(bar) and let T ϵ L(V,W). A linear map T* ϵ L(W,V) such that < Tv, w > = < v, T*w > (v ϵ V, w ϵ W) is called the adjoint of T.
The adjointt of a linear map is unique when it exists.

Question 5

Definition 3.3 : Normal , Hemitian and unitary, maps

Answer 5

Let V be an inner product space over C(bar). A map T ϵ L(V,W) for which T* exists is called
- NORMAL if T* T =T T*
- HERMITIAN (self adjoint) if T* = T
- UNITARY if T T=id=T T
Note: every hermitian and every unitary map is normal .

Question 6

Definition 3.5 : Normal , Hemitian and unitary, matrices

Answer 6

A matrix A ϵ M_n(Cbar) is called
- NORMAL if A^A = AA^
- HERMITIAN (self adjoint) if A^=A
- UNITARY if A^A=id=A^A*
Note: every hermitian and every unitary matrix is normal .

Question 7

Theorem 3.6 : Adjoints - Linear maps and matrices

Answer 7

Let V be a finite-dimensional inner product space over C(bar) and let μ and υ be orthonormal bases of it. Let T ϵ L(V) be a linear map.
Then,
1) Φ_υ,μ (T) = Φ_μ,υ (T)
2) there exists a unitary matrix U such that Φ_μ (T) = U*Φ_υ (T)U
3) T is normal if and only if Φ_μ (T) is normal.
4)T is hermitian if and only if Φ_μ (T) is hermitian.
5) T is unitary if and only if Φ_μ (T) is unitary.

Question 8

Theorem 3.8 : Spectral theorem Complex case

Answer 8

Let V be a finite- dimensional inner product space over C(bar) and let T ϵ L(V). Then there exists an orthonormal basis μ such that Φ_μ (T) is diagonal if and only if T is normal.

Question 9

Theorem 4.1 : Spectral theorem Real case

Answer 9

Let V be a finite- dimensional inner product space over R(bar) and let T ϵ L(V). Then there exists an orthonormal basis μ such that Φ_μ (T) is diagonal if and only if T is symmetric.

In the real case : unitary is replaced by orthogonal and hermitian is replaced by symmetric.