Inner Products

Question 1

Definition of a real inner product

Answer 1

Symmetry:
(u,v) = (v,u)
for complex (u,v) = conjugate(v,u)

Linearity:
(u+v,w) = (u,w) + (v,w)

(λu,v) = λ(u,v)

Positivity:
(u,u) is greater than or equal to 0, with equality iff u = 0

Same holds for complex inner products

Question 2

Norm of v =

Answer 2

sqrt(v,v)

Question 3

If u and v are orthogonal

||u + v||^2 =

Answer 3

||u||^2 + ||v||^2

Question 4

A* =

Answer 4

Conjugate of A^t

Question 5

A matrix is hermitian if…

Answer 5

A = A*

Question 6

A matrix is anti-hermitian if…

Answer 6

A* = -A

Question 7

Complex Cauchy-Schwarz inequality

Answer 7

|⟨u,v⟩|^2 ≤ ||u||^2 · ||v||^2
with equality iff u and v are linearly dependent
also true for real inner products

Question 8

G.S decomposition

Answer 8

Take e.g. 3 vectors v1, v2, v3
u1 = v1 / ||v1||

u2 = ṽ2 / ||ṽ2||
ṽ2 = v2 - (v2,u1)u1

u3 = ṽ3 / ||ṽ3||
ṽ3 = v3 - (v3,u1)u1 - (v3,u2)u2

Question 9

Unitary matrix

Answer 9

MM = MM = I

Question 10

Special unitary group SU(n) =

Answer 10

{M unitary with determinant +/- 1}

Question 11

How to find e.g. h in S closest to f(t) = t^3

Answer 11

Basis of S = {f1, f2, f3}
Find orthonormal basis using G.S g1, g2, g3

h = Ps(f) = (f, g1)g1 + (f,g2)g2 + (f,g3)g3