QR Decomposition And Orthoganality

Question 1

What does QR decomposition theorem state

Answer 1

QR decomposition theorem states that:
If A is a square invertible matrix, then we can write
A = QR where Q is orthogonal and R is upper triangular with positive entries on diagonal (Rij = 0 if i > j)

Question 2

How to compute Q and R in practice

Answer 2

To compute Q and R in practice:

Do Gram-Schmidt orthogonalisation on columns of A to get Q and then R = Q^T *A

Question 3

What is definition of orthogonal complement

Answer 3

Orthogonal complement definition is:
Orthogonal complement U perp of U <= V is given by:
U perp = {v in V : inner(u,v) = 0 for all u in U)
(perpendicular lines in R^2)

Question 4

If V = U + U perp, then what is V isomorphic to

Answer 4

If V = U + U perp, then V is isomorphic to U direct sum +o U perp

Question 5

If U + U perp = V, then what is dim U perp and (U perp) perp

Answer 5

If U + U perp = V, then dim U perp = dim V - Dim U

(U perp) perp = U