Description

General Instructions
The following document contains the solutions to the theory-based questions for
Question 1
Finding the singular values of
ATA
− λI| = 10 200 − λ 120
λ = 332.17,90.50,0.33
Thus, the largest and smallest singular values of the matrix are and
Similarly for B
BTB
− λI| = 40 800 − λ 480
λ = 1328.68,361.99,1.33
Thus, the largest and smallest singular values of the matrix are and

Note that B = 2A and the eigenvalues of B are 4 times the eigenvalues of A.
Thus we can conclude that κ(αA) = κ(A) but under the added constraint of α ∈ R∖{0}
Question 2

The transformation matrix A corresponding to the rotation of a vector x by an angle θ is given by
cosθ −sinθ
A = [ ] sinθ cosθ
Computing the singular values of A by computing the eigenvalues of ATA
T cosθ sinθ cosθ −sinθ
A A = [ ][ ]
−sinθ cosθ sinθ cosθ cos2 θ + sin2 θ −cosθsinθ + sinθcosθ
=−sinθcosθ + cosθsinθ sin2 θ + cos2 θ ]
1 0
= [ ]
0 1
The eigenvalues of ATA are λ1 = 1 and λ2 = 1. Thus, the singular values of A are σ1 = 1 and σ2 = 1.
Thus, the condition number of A is given by
κ(A) = ||σσ12|| = 11 = 1
Question 3

Finding the transformation matrix A corresponding to the reflection of a vector x about a given vector v.
Let θ be the angle between x and v given by .
For reflection about v, we need to rotate x by 2θ. Thus, the transformation matrix A is given by
cos2θ
A = [
sin2θ −sin2θ] cos2θ
As we calculated in the previous question the singular values of the above matrix are σ1 = 1 and σ2 = 1 (as they don’t depend on θ) Thus, the condition number of A is given by
κ(A) = ||σσ12|| = 11 = 1