Properties of condition numbers-orthogonal matrices

Question

Problems:

Properties of condition numbers : Orthogonal Matrices and Eigenvalues

1. k(λA) = k(A) for λ≠0

2. k(A) = [max_||x||=1 ||Ax||]/[min_||x||=1 ||Ax||]

3. If a_j denotes the j-th column of A, then k(A) > ||a_j||/||a_j||

4. k₂(A) = k₂(A^t)

5. k(I) = 1

6. k(A) > 1

7. For any orthogonal matrix Q,

                           k₂(QA) = k₂(AQ) = k₂(A)

8. If D = diag(d₁,...,d_n) then

                         k2(D) = k1 (D) = k∞(D) = [max_1<i<n |di|]/[min_1<i<n |di|]

9. If λ_M is the largest eigenvalue of A^tA and λ_m is its smallest eigenvalue, then

              k₂(A^tA) = k₂(AA^t) = (k₂(A))² = λ_M/λ_m

10. k2(A) = 1 if and only if A is a multiple of an orthogonal matrix.