Singular Value Decomposition

Question 1

If phi is a linear operator in a finite dimensional I.P space, what are properties of phi

Answer 1

If phi is a linear operator in a finite dimensional I.P space, what are properties of phi are:
All eigenvalues of phi o phi* are non-negative
Ker(phi* o phi) = ker(phi)

Question 2

If phi is a linear operator, what are singular values of phi

Answer 2

If phi is a linear operator, singular values of phi are:

Sigma(1),…,sigma(n) where sigma(i) = root(mu(i)) where mu(1),…,mu(n) are eigenvalues of phi* o phi with multiplicity

Question 3

What does singular value decomposition theorem state

Answer 3

Singular value decomposition theorem states that:
If V is finite dimensional I.P space and phi is a linear operator with singular values simga(1),…,sigma(n) then there are orthonormal basis u1,…,uN and w1,…,wn of V s.t
Phi(v) = sum i=1 to n sigma(i) * inner(u(i), v) * w(i)