Adjoints And Matrices

Question 1

What does V have if it is a finite dimensional inner product space

Answer 1

V has an orthonormalisation basis is V is a finite dimensional inner product space

Question 2

When is phi* adjoint of phi (set of linear operators V to V)

Answer 2

Phi* is adjoint of phi if :
Inner(phi(v),w)= inner(v,phi(w)) or equivalent,y by conjugate symmetry
Inner(w, phi(v))= inner(phi(w), v) for all v,w E V

Question 3

If phi is represented by matrix A then what is phi* represented by

Answer 3

If phi is represented by matrix A, then phi* is represented by complex conjugate A^T

Question 4

When is phi self adjoint or skew adjoint

Answer 4

Phi is self adjoint when phi=phi* eq inner(phi(v),w) = inner(v,phi(w))
Skew-adjoint of phi* = -phi eq inner(phi(v),w) = -inner(v,phi(w))

Question 5

When is psi (V to W) a linear isometry where V and W are I.P spaces

Answer 5

Psi is a linear isometry if:

Inner (phi(v1),phi(v2) in W = inner(v1,v2) in V

Question 6

What is an orthogonal transformation phi (is an isomorphism)

Answer 6

An orthogonal transformation phi is when:

Phi^-1 = phi* if field = R

Question 7

What is unitary transformation

Answer 7

Unitary transformation is when:

Phi^-1 = phi* if field = C

Question 8

When is matrix A orthogonal

Answer 8

Matrix A is orthogonal if field = R and A^-1 =A^T

Question 9

When is matrix A unitary

Answer 9

Matrix A is unitary if field = C and A^-1 = A(dagger)

Question 10

What is rigid motion

Answer 10

Rigid motion (or distance preserving) is a map f (V to V) if mod(f(v) - f(w)) = mod(v-w) for all v,w E V