Lecture 4: Linear Operators on Vector Spaces

Question 1

Linear Operator Matrix y = L(x), B = Aa

Answer 1

1. Linear operator satisfies superposition
2. Relation between input and output representations
3. Used to find other relations

Question 2

Properties of Representation Matrix A

Answer 2

ith Column of A is the representation of yi = L(ei) with respect to output basis

Question 3

Progression of Linear Operator Matrix Equations

1. input as matrix form
2. output as matrix form
3. output columns

Answer 3

1. x = [e1 … en] * [a1;…;an]
2. y = L(x) = a1*L(e1) + … + anL(en) = UAa
3. yi = L(ei) = [e1 … en] * ai

Question 4

Relationship between two representations and basis of x

Answer 4

1. a’ = Pa

2. ith column of P is the representation of original basis vector ei with new basis vectors e’

Question 5

Relationship between two representations and basis of y

Answer 5

1. B’ = TB

2. ith column of T is the representation of original basis vector ui with new basis vectors u’

Question 6

Relationship Between Matrix Representations

Answer 6

A = inverse(T) * A' * P
A' = T * A * inverse(P)

Question 7

Similarity Transformation

Answer 7

A and A’ are similar if there exists a nonsingular matrix P satisfying
A = inverse(P) * A’ * P
A’ = P * A * inverse(P)

Question 8

Operators as Spaces

Answer 8

1. Set of all linear operators is a linear space
2. Follows axioms of Norms
3. Norm used to find size of operator

Question 9

Matrix Norms

1. Supremum
2. A1 Norm
3. Euclidean Norm
4. Infinity Norm
5. Frobenius Norm

Answer 9

1. Least upper bound
2. Largest sum across columns
3. sqrt(Max(x’T *A’T Ax))
4. Largest sum across rows
5. sqrt(tr(A’T *A)

Question 10

Bounded Linear Operator

Answer 10

||Ax|| <= k ||x|| for all vectors x in space.

If gain ||Ax|| / ||x|| is bounded

Question 11

Adjoint Operator L*

Answer 11

Must satisfy = for all vectors x and y (<> denotes inner product)
ie. A represents operator L where an orthonormal basis to obtain matrix representation A, then Adjoint of L is transpose(A’).

Question 12

Self-Adjoint Operators

Answer 12

If adjoint operator is the same as the operator to begin with it is called the hermitian operator. Operator must have orthonormal basis.
If hermitian is real then A = transpose(A).